REVIEW OF ESSENTIAL MATH 346 TOPICS

Size: px
Start display at page:

Download "REVIEW OF ESSENTIAL MATH 346 TOPICS"

Transcription

1 REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations and the order relation < satisfying the following axioms: (a) (R, +,.) is a field with additive identity 0 and multiplicative identity 1. (b) (R, <) is a partially ordered set compatible with the field axioms in the sense that (i) if x < y and y < z, then x < z, (ii) if x < y, then x + z < y + z for all z R, (iii) if x > 0 and y > 0, then xy > 0, and (iv) x, y R, either x > y, or y > x, or x = y. [Trichotomy] (c) (Completeness axiom) Every nonempty set of real numbers bounded above has a least upper bound (supremum). [Alternative statement: Every nonempty set of real numbers bounded below has a greatest lower bound (infimum).] Remarks. 1. Any complete ordered field is isomorphic to (R, +,., ) with Completeness axiom. 2. For a set S R, sup(s) need not belong to S. 3. If sup(s) exists, it is unique. (Hence we have set functions!) 4. Since inf( S) = sup(s), corresponding statements also hold for inf. 5. R is unbounded (hence so are N, Z and Q). For any x, y R, define the absolute value of x by x = interpreted as the distance of x to 0. For any x, y R, define maximum and minimum of x and y by { { x if x y y if x y x y = and x y = y if otherwise x if otherwise, { x if x 0 x if x < 0, respectively. which can be Exercises. 1. If x + ɛ y for all ɛ > 0, then x y. 2. Let A R be nonempty and α R be an upper bound for A. α = sup(a) iff ɛ > 0 x A such that α ɛ < x α. 3. If a < c for all c with c > b, then a b. 4. If a b a A, then sup(a) b. 5. If A R is bounded (i.e. bounded above and below) and B A is nonempty, then inf(a) inf(b) sup(b) sup(a). 6. Let A and B be nonempty subsets of R such that a b a A b B. Then (i) sup(a) sup(b) if sup(b) exists, and (ii) sup(a) inf(b). 1

2 2 7. Let A be a nonempty subset of R with α = sup(a). If for c 0, ca := {ca : a A}, then cα = sup(ca). 8. Let A and B be nonempty subsets of R and let C = {a + b : a A, b B}. If sup(a) and sup(b) exist, then so does sup(c) and sup(c) = sup(a) + sup(b). 9. Exercises 2, 4, 6-8 are also is true for infima with obvious (appropriate) changes in the statements. Fact. The following statements are a consequence of the axioms (a)-(c) above: (i) x R n Z + such that n > x. (ii) (Archimedean Property) If x, y R with x 0, then n Z such that y < nx. (iii) If x, y R with x < y, then z Q (Q c ) such that x < z < y. (iv) x > 0 n Z + such that n 1 x < n. v) ɛ > 0 n Z + such that 1 n < ɛ. 2. THE EXTENDED REAL NUMBER SYSTEM The set R # := R { } with operations (i) x + = for all x R; x( ) = if x > 0 and ( x)( ) = ± (ii) + =, ( ) + ( ) = ;.( ) = ; 0.( ) = 0 and with the order property that < x < for all x R, is called the extended real number system. Remarks. 1) is not defined. 2) sup( ) = is assumed. 3) If a set A of real numbers has no upper bound, then we say sup(a) =. Hence, in R # every set has a supremum (infimum). 3. CARDINALITY Let A and B be two nonempty sets. We say that A is equipotent with, or equinumerous with, or similar to B, denoted by A B, if there exists f : A B which is 1-1 and onto. Remark. The relation is an equivalence relation on the family of sets A set A is called a finite set if there exists n Z + such that A {1, 2,..., n}. In this case we say that n is the cardinal number of A, and denote it by A = n. Convention: = 0. A set which is not finite is called an infinite set. Such a set A is called countably infinite if A Z +. Notation. We denote the cardinal number of Z + by ℵ 0. Hence Z + = ℵ 0. Observe that N Z + ; countably infinite. Z Z +. Hence, we have that N = Z = ℵ 0 ; i.e., both N and Z are

3 Second observation: The process of counting is essentially the process of determining cardinality of finite sets. Extending this idea to countably infinite sets, we see that a set is countably infinite if we can list (or label) the elements of A as a sequence a 1, a 2, a 3,..., a n, A set which is finite or countably infinite is called a countable set or a denumerable set. If a set is not countable, it is called an uncountable, or non-denumerable set. Examples. 1. Z +, N, Z are countable. 2. Since Z + 2Z + = {2n : n Z + } (why?), it follows that 2Z + is also countable. Remark. The example 2 above manifests an important characteristics of infinite sets; namely, an infinite set is the one that is similar to a proper subset of itself. Facts. a) If A is a countable, then every B A is also countable. b) The union of countably many countable sets is a countable set. Examples. 1. The sets D = {2 n 5 m ; n, m N}, Z Z and the set of all prime numbers are countable. 2. Q and Q Q are countable. 3. The set s = {(a, b) : a, b Q, a < b} is countable. 4. The set of all polynomials with rational coefficients is countable. Consequently, the set of algebraic numbers is countable. Question: Are there uncountable sets? The answer is Yes! Indeed, many! Fact. The set of points in the interval (0, 1) is uncountable. Corollary. R is countable. Notation. We denote the cardinal number of the set (0, 1) by c. Hence (0, 1) = R = c. Other examples of uncountable sets are: Q c. Any nonempty (open, closed, or half-open) interval. C. {0, 1} N = {(a n ) : a n {0, 1} n N}. R R. The set of all transcendental numbers in R. Remark. ℵ 0 <? < c! 4. THE TOPOLOGY OF R For any a R the (open) ball centered at a with radius r > 0 is the set B(a, r) := {x R : x a < r} = (a r, a + r). For a set A R, a point a A is said to be an interior point if there exists ɛ > 0 such that B(a, ɛ) A. The set of all interior points of a set A is called the interior of A and is denoted by int(a). A set A R is called open if every point of it is an interior point. A set A R is called closed if A c is open. The empty set is assumed open (and closed).

4 4 Facts. a) The union (intersection) of any collection of open (closed) sets is open (closed). b) The union (intersection) of any finite collection of closed (open) sets is closed (open). Fact. Every nonempty open set of real numbers is a disjoint countable union of open intervals. A real number p is called an accumulation point (or cluster point, or limit point) of a set A R if for every ɛ > 0 we have B(p, ɛ) A. The set of all accumulation points of a set A is called the derived set of A and is denoted by A. Theorem. point. (Bolzano-Weierstrass) Every bounded infinite subset of R has an accumulation The smallest closed set containing a set A is called the closure of A and is denoted by A. Facts. Let A R be a (nonempty) subset. Then a) int(a) A Ā. b) A is closed iff A = Ā. c) int(a) is the largest open set contained in A, and A is the smallest set containing A. d) Q = R. (Hence, R is separable.) Exercises. Let A, B R, then: 1. Ā = A A. 2. int(a) int(b) int(a B) and int(a) int(b) = int(a B). 3. Ā B = A B and A B Ā B. Theorem. (Nested Set Property or Cantor Intersection Theorem) If {F n } is a collection of closed and bounded set of real numbers such that F n F n 1 for all n 1, then n 1 F n. A collection U = {U λ : λ Λ} of subsets of R is called a cover for A R if A λ Λ U λ. The collection U is called an open cover for A if each U λ is open. Theorem. (Heine-Borel) Let A R be a closed and bounded set and U be an open cover. Then there is a finite subcollection of U that covers A. A set C R is called a compact set if every open cover of it has a subcover consisting of finitely many elements. Corollary The following statements are equivalent: a) A R is compact. b) A is closed and bounded. c) Every infinite subset of A has an accumulation point in A. 5. METRIC SPACES A metric space is a pair (M, d), where M is a nonempty set and d : M M R is a mapping that satisfies (i) d(x, y) 0 for all x, y M, (ii) d(x, y) = 0 x = y, (iii) d(x, y) = d(y, x) for all x, y M, and

5 5 (iv) (Triangle inequality) d(x, y) d(x, z) + d(z, y) for all x, y, z M. Examples. 1. On R, the map d(x, y) = x y defines a metric; hence (R, d) is a metric space. The topology of R studied in the previous section is given by this metric. 2. For x = (x i ), y = (y i ) R n (or C n ), the map d 2 (x, y) = ( n i=1 x i y i 2 ) 1 2, defines a metric on R n (or C n ), called the Euclidean metric, making (R n, d 2 ) (or (C n, d 2 )) a metric space. 3. For x = (x i ), y = (y i ) R n, let d 1 (x, y) = n i=1 x i y i and d (x, y) = max 1 i n x i y i. Then both d 1 and d define metrics on R n. So (R n, d 1 ) and (R n, d ) are metric spaces. 4. Let M = [ π, π ]. Then the map d(x, y) = tan x tan y is a metric on M; hence, (M, d) 2 2 is a metric space. 5. The map ρ(z, w) = z w 1+ z w defines a metric on C; hence, (C, ρ) is a metric space. Given an inner product space (V, <, >), the the inner product naturally defines a metric on V by d(x, y) =< x y, x y > 1 2 = x y, for x, y V, since, by the properties of inner product listed above, (i) d(x, y) = x y 0 for all x, y V ; and d(x, y) = x y = 0 x = y, (ii) d(x, y) = x y = y x = d(y, x) for all x, y V, and (iii) d(x, y) = x y x z + z y = d(x, z) + d(z, y) for all x, y, z V. Hence, on an inner product space the vector space and metric space structure are compatible. Observe that, the metric in the Example 2 above is actually the metric on R n induced by the usual inner product on R n (or C n ). Let (M, d) be a metric space, then for any a M and any real number r > 0, the set B(a, r) := {x M : d(x, a) < r} is called the (open) ball centered at a with radius r. An element a M is an interior point of a set A M if there exists ɛ > 0 such that B(a, ɛ) A, and the set of all interior points of A is called the interior of A, denoted by int(a). A set A M is open if A = int(a) and it is closed if A c is open. The empty set is assumed open (and hence, closed). A point a M is called an accumulation point (or a limit point) of a set A M if for every ɛ > 0, (B(a, ɛ) \ {a}) A. The set of all accumulation points of A is called the derived set of A and is denoted by A. The smallest closed set containing a set A M is called the closure of A and is denoted by Ā. Fact. Let (M, d) be a metric space and A, B M. Then: a) int(a) A A. b) A is closed iff A = A. c) Ā = A A. d) int(a) is the largest open set contained in A, and A is the smallest set that contains A. e) The union (intersection) of any collection of open (closed) sets is open (closed). f) The union (intersection) of any finite collection of closed (open) sets is closed (open). g) int(a) int(b) int(a B) and int(a) int(b) = int(a B). h) Ā B = A B and A B Ā B. A metric space M is called separable if there is a countable subset A M such that M = Ā.

6 6 A collection U = {U λ : λ Λ} of subsets of M is called a cover for A M if A λ Λ U λ. The collection U is called an open cover for A if each U λ is open. A set C M is called a compact set if every open cover of it has a subcover consisting of finitely many elements. For example, any closed interval [a, b] R, where < a < b <, is a compact set; on the other hand, [a, b) is not a compact set. Fact. If A M is compact, then a) A is closed and bounded. b) Every infinite subset of A has an accumulation point in A. It follows that any closed subset of a compact set is itself compact. compact metric space is separable. Furthermore, every A sequence in M is a function a : N M. For convenience we denote sequences by (a n ) n 1, where a(n) = a n, n 1. A sequence (a n ) n 1 is called bounded if there exists a real number r > 0 such that {a n } n B(a, r), for some a M. By definition, a sequence is an infinite set of points in M; hence, if bounded, it has at least one accumulation point, say a, by Fact 2(b). If a sequence has only one accumulation point, it is called the limit of the sequence and the sequence is called convergent or we say that the sequence converges to a, which is denoted by a n a or lim n a n = a. More explicitly, a n a if and only if ɛ > 0 N Z + such that n N d(a n, a) < ɛ. Fact. Let A M. Then a A if there exists (a n ) A such that a n a. One can characterize compactness in terms of sequences: Fact. A set A M is compact if and only if every sequence {a k } A has a subsequence that converges to an element in A. A sequence (a n ) M is called a Cauchy sequence if ɛ > 0 N Z + such that m, n N d(a n, a m ) < ɛ. In a metric space M, every convergent sequence is Cauchy. Furthermore, if a sequence (a n ) M is Cauchy, then it is bounded. On the other hand, there are metric spaces in which some Cauchy sequences are not convergent. A metric space (M, d) is called complete if every Cauchy sequence in M is a convergent sequence. From the characterizations above, it follows that every compact metric space is complete. Clearly, not every complete metric space is compact; however, the compact ones can still be characterized. For, a set A M is called totally bounded if, ɛ > 0, a finite set {x 1,..., x n } M, which is called an ɛ-net, such that A n k=1 B(x k, ɛ). Theorem. Let (M, d) be a metric space. a) A totally bounded set A M is bounded and separable. b) The closure A of a totally bounded set A M is totally bounded. c) If M is complete, a subset of M is compact if and only if it is totally bounded and closed.

7 7 6. SEQUENCES OF REAL NUMBERS Recall that a sequence is a function a : N R; for convenience, we denote sequences by (a n ) n 1, where a(n) = a n, n 1. By definition, a sequence is an infinite set of real numbers; hence, if bounded, it has at least one accumulation point, say a, by Bolzano-Weierstrass Theorem. Hence, we have Theorem. (Bolzano-Weierstrass Thoerem for sequences, version 1) Every bounded sequence of real numbers has an accumulation point. If a sequence has only one accumulation point, it is called the limit of the sequence and the sequence is called convergent or we say that the sequence converges to a, and is denoted by a n a or lim n a n = a. More explicitly, a n a if and only if ɛ > 0 N Z + such that n N a n a < ɛ. An equivalent statement for a n a in terms of open balls is as follows: a n a if and only if ɛ > 0 N Z + such that, whenever n N then a n B(a, ɛ). Fact. Every convergent sequence of real numbers is bounded (i.e., M > 0 such that a n M for all n). Remark. Converse of the previous statement is not valid. Fact. Let A R. Then a A if there exists (a n ) A such that a n a. Remark. The point a in the Fact above need not be in A. A sequence (a n ) is called a Cauchy sequence if ɛ > 0 N Z + such that m, n N a n a m < ɛ. Exercise. If (a n ) is a convergent (Cauchy) sequence, then it is bounded. Remarks. 1) lim n a n = means that c > 0 N > 0 such that n N a n > c. Similarly, lim n a n = means that c < 0 N > 0 such that n N a n < c. 2) In general, if we say a n a, then < a < ; if (a n ) R #, then a. Observe that a n a means, for any ɛ > 0, all but finitely many a n s are in the interval (a ɛ, a + ɛ). A weakening of this is requiring infinitely many of a n s are in (a ɛ, a + ɛ). Fact. a R is an accumulation point (or cluster point) of a sequence (a n ) iff ɛ > 0 there exists infinitely many a n (a ɛ, a + ɛ). Equivalently, a is a cluster point of (a n ) iff ɛ > 0 and m Z +, n m such that a n (a ɛ, a + ɛ). Examples. 1) a n : 1, 1, 1/2, 1, 1/3, 1, 1/4,.... Accumulation points are 0, 1. 2) (( 1) n ). Accumulation points are -1, 1. 3) ( 1 ln n ) n=2. Accumulation point is 0. Remark. If a sequence (a n ) has more than one accumulation points, then it is not a convergent sequence; however, for each accumulation point, it has a subsequence convergent to that

8 8 accumulation point (Exercise). In particular, if a n a, then a is an accumulation point (the only one). Fact. (Bolzano-Weierstrass Theorem for sequences, version 2) Every bounded sequence of real numbers has a convergent subsequence. Question. How do we know that a given sequence has a limit? Fact. Every bounded monotone sequence of real numbers is convergent. Proof. (Sketch) By Bolzano-Weierstrass Thoerem for sequences, version 1, the sequence has an accumulation point. Since it s monotone, it has only one accumulation point; hence, it must be convergent. Fact. (Cauchy Criterion for sequences) A sequence of real numbers (a n ) is convergent if an only if it is a Cauchy sequence. Fact. (Algebra of limits) Let (a n ) and (b n ) be sequences and α R, such that a n a, b n b. Then (i) a n + b n a + b and a n b n ab (ii) αa n αa (iii) an b n ab, provided that b 0 Fact. (Squeeze Theorem) Let a n c n b n for all n 1. If a n a and b n a, then c n a. Question. Can we associate a real number to any (not necessarily convergent) sequence of real numbers? First, recall that supremum and infimum of any bounded set of real numbers exit (if unbounded, they exist in R # ). Now, given any sequence of real numbers (a n ), define, for k 1, a k = inf{a k, a k+1, a k+2,... } and a k = sup{a k, a k+1, a k+2,... }. Then, it follows that a k a k+1 and a k a k+1 for all k 1. Hence, {a k } is monotone increasing and {a k } is monotone decreasing sequence. Therefore, lim k a k and lim k a k exist (in R # ). (If (a n ) is bounded, then lim k a k and lim k a k exist in R.) Also, observe that lim k a k = sup k 1 inf n k {a n }, and lim k a k = inf k 1 sup n k {a n }. Definition. For any sequence of real numbers (a n ), define lim sup n a n and lim inf n a n as lim sup a n = lima n = inf n n lim inf n sup k 1 n k {a n }, and a n = lim n a n = sup inf {a n}. k 1 n k Remarks. 1. For all n 1, a k a n a k by construction. 2. For any i, j 1, we have a i a i+j a i+j a i ; hence, lim inf n a n lim sup n a n.

9 9 3. lim inf n a n = lim sup n a n if and only if (a n ) is convergent; in that case, lim inf n 4. lim inf n ( a n ) = lim sup n a n. a n = lim sup a n = lim a n. n Exercise. Prove that lim inf n a n (lim sup n a n ) is the smallest (largest) of all the limit points of the set {a n }. Given a sequence of real numbers (a n ) n=1, we can study the infinite series k=1 a k using the tools developed for sequences. Let (s n ) n=1a k be the sequence of partial sums of the series k=1 a k, where s n = n k=1 a k. We say that the infinite series k=1 a k is convergent if s n s, for some real number s. In this case we write k=1 a k = s and call s as the sum of the series. If a series is not convergent, it is called divergent. From the definition we have: Fact. k=1 a k is convergent iff ɛ > 0, N Z + such that n N = k=n+1 a k < ɛ. Fact. (Cauchy Criterion for series) k=1 a k is convergent iff ɛ > 0, N Z + such that m, n N = n k=m+1 a k < ɛ. Fact. (Algebra of series) Let k=1 a k and k=1 b k be convergent series with sums a and b, respectively. Then (i) k=1 a k + k=1 b k = a + b (ii) α k=1 a k = αa, α R Tests for convergence of series: 1. (Divergence Test) a n 0 = k=1 a k is divergent. 2. (Comparision Test) If 0 a n b n, then (i) k=1 a k divergent = k=1 b k divergent. (ii) k=1 b k convergent = k=1 a k convergent. k=1 a k. 4. (Alternating Series Test) If (a n ) is a sequence such that (i) a n a n+1, forall n 1, and 3. If k=1 a k is convergent, then so is (ii) a n 0, then k=1 ( 1)k+1 a k is convergent. 7. REAL-VALUED FUNCTIONS Let A R, a A, and f : A R be a function. We say that f has limit L at a, denoted by lim x a f(x) = L, iff ɛ > 0, δ > 0 such that x B(a, δ) \ {a} f(x) B(L, ɛ). Fact. lim x a f(x) = L, iff sequence (a n ) A with a n a, we have lim n f(x) = L. We recall the following properties of limits: Let f, g : A R be two functions such that lim x a f(x) = L and lim x a g(x) = M, where a A. Then a) lim x a (f + g)(x) = L + M and lim x a (fg)(x) = LM, b) lim x a αf(x) = αl, α R, f(x) c) lim x a = L, provided that M 0. g(x) M

10 10 Let A R and a A. Recall that a function f : A R is continuous at a iff ɛ > 0, δ > 0 such that if x B(a, δ) then f(x) B(f(a), ɛ). Equivalently, f is continuous at a A iff ɛ > 0, δ > 0 such that x a < δ = f(x) f(a) < ɛ. Fact. A function f : A R is continuous at a A if and only if for any sequence (a n ) A with a n a, f(a n ) f(a). Let f, g : A R be functions continuous at a A. Then a) f + g and fg are continuous at a. b) α R, αf is continuous at a. c) f f(x) is continuous at a, provided that is defined for all x B(a, ɛ), for some ɛ > 0. g g(x) If f : A R is continuous at a A, and g : f(a) R is continuous at f(a), then f g : A R is continuous at a. Fact. A function f : A R is continuous on A if and only if for any open set O R, f 1 (O) is (relatively) open in A. The following theorems indicate the reason why we value continuous functions. Theorem. (Extreme Value Theorem) Every continuous function f : A R, where A R is compact, attains both of its extrema. Theorem. (Intermediate Value Theorem) If f : [a, b] R is continuous, where < a < b <, and f(a) < γ < f(b), then c (a, b) such that f(c) = γ. Theorem. Let f : A R be a continuous function on A and let B A be a compact (connected) subset. Then f(b) is compact (connected). Remark. It is not always true that if A is open (closed), then so is its forward image f(a). Recall that a function f : A R is uniformly continuous on A if and only if ɛ > 0, δ > 0 such that if x y < δ then f(x) f(y) < ɛ for all x, y A. Note that not every continuous function is uniformly continuous; however, under some conditions this is true. Theorem. If f : A R is continuous (on A) and A is compact, then f is uniformly continuous (on A). Given a function f : A R and a A. Then f fails to be continuous at a if (i) lim x a f(x) exists but not equal to f(a) (removable discontinuity), or (ii) lim x a + f(x) lim x a f(x) (jump discontinuity), or (iii) lim x a f(x) does not exist (essential discontinuity). In general, it is rather complicated to describe the set of points of discontinuity for a given function. In the case of monotonic functions, however, we can achieve this task. Fact. Let f : A R be a monotonic function. Then (i) All the discontinuities of f are jump discontinuity. (ii) If D f A denotes the set of points of discontinuity of f, then D f is at most countable.

Logical Connectives and Quantifiers

Logical Connectives and Quantifiers Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then

More information

Problem Set 2: Solutions Math 201A: Fall 2016

Problem Set 2: Solutions Math 201A: Fall 2016 Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that

More information

Some Background Material

Some Background Material Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

More information

A LITTLE REAL ANALYSIS AND TOPOLOGY

A LITTLE REAL ANALYSIS AND TOPOLOGY A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3 Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

More information

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered

More information

11691 Review Guideline Real Analysis. Real Analysis. - According to Principles of Mathematical Analysis by Walter Rudin (Chapter 1-4)

11691 Review Guideline Real Analysis. Real Analysis. - According to Principles of Mathematical Analysis by Walter Rudin (Chapter 1-4) Real Analysis - According to Principles of Mathematical Analysis by Walter Rudin (Chapter 1-4) 1 The Real and Complex Number Set: a collection of objects. Proper subset: if A B, then call A a proper subset

More information

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

More information

Exercises from other sources REAL NUMBERS 2,...,

Exercises from other sources REAL NUMBERS 2,..., Exercises from other sources REAL NUMBERS 1. Find the supremum and infimum of the following sets: a) {1, b) c) 12, 13, 14, }, { 1 3, 4 9, 13 27, 40 } 81,, { 2, 2 + 2, 2 + 2 + } 2,..., d) {n N : n 2 < 10},

More information

Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3

Contents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3 Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability

More information

That is, there is an element

That is, there is an element Section 3.1: Mathematical Induction Let N denote the set of natural numbers (positive integers). N = {1, 2, 3, 4, } Axiom: If S is a nonempty subset of N, then S has a least element. That is, there is

More information

Metric Spaces Math 413 Honors Project

Metric Spaces Math 413 Honors Project Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only

More information

Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

Definitions & Theorems

Definitions & Theorems Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic 2 1.1 Sets.................................................. 2 1.2 The Peano axioms..........................................

More information

Theorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers

Theorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers Page 1 Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property Suppose is an ordered set with the least-upper-bound property Suppose, and is bounded below be the set of lower

More information

Real Analysis - Notes and After Notes Fall 2008

Real Analysis - Notes and After Notes Fall 2008 Real Analysis - Notes and After Notes Fall 2008 October 29, 2008 1 Introduction into proof August 20, 2008 First we will go through some simple proofs to learn how one writes a rigorous proof. Let start

More information

Metric Spaces Math 413 Honors Project

Metric Spaces Math 413 Honors Project Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only

More information

Copyright c 2007 Jason Underdown Some rights reserved. statement. sentential connectives. negation. conjunction. disjunction

Copyright c 2007 Jason Underdown Some rights reserved. statement. sentential connectives. negation. conjunction. disjunction Copyright & License Copyright c 2007 Jason Underdown Some rights reserved. statement sentential connectives negation conjunction disjunction implication or conditional antecedant & consequent hypothesis

More information

Metric Spaces and Topology

Metric Spaces and Topology Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

More information

Chapter 2 Metric Spaces

Chapter 2 Metric Spaces Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics

More information

MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE

MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE MATH 101, FALL 2018: SUPPLEMENTARY NOTES ON THE REAL LINE SEBASTIEN VASEY These notes describe the material for November 26, 2018 (while similar content is in Abbott s book, the presentation here is different).

More information

4 Countability axioms

4 Countability axioms 4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

ANALYSIS WORKSHEET II: METRIC SPACES

ANALYSIS WORKSHEET II: METRIC SPACES ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair

More information

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

More information

Real Analysis. Joe Patten August 12, 2018

Real Analysis. Joe Patten August 12, 2018 Real Analysis Joe Patten August 12, 2018 1 Relations and Functions 1.1 Relations A (binary) relation, R, from set A to set B is a subset of A B. Since R is a subset of A B, it is a set of ordered pairs.

More information

Essential Background for Real Analysis I (MATH 5210)

Essential Background for Real Analysis I (MATH 5210) Background Material 1 Essential Background for Real Analysis I (MATH 5210) Note. These notes contain several definitions, theorems, and examples from Analysis I (MATH 4217/5217) which you must know for

More information

Lecture 2: A crash course in Real Analysis

Lecture 2: A crash course in Real Analysis EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 2: A crash course in Real Analysis Lecturer: Dr. Krishna Jagannathan Scribe: Sudharsan Parthasarathy This lecture is

More information

FUNDAMENTALS OF REAL ANALYSIS by. II.1. Prelude. Recall that the Riemann integral of a real-valued function f on an interval [a, b] is defined as

FUNDAMENTALS OF REAL ANALYSIS by. II.1. Prelude. Recall that the Riemann integral of a real-valued function f on an interval [a, b] is defined as FUNDAMENTALS OF REAL ANALYSIS by Doğan Çömez II. MEASURES AND MEASURE SPACES II.1. Prelude Recall that the Riemann integral of a real-valued function f on an interval [a, b] is defined as b n f(xdx :=

More information

Chapter 1 The Real Numbers

Chapter 1 The Real Numbers Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus

More information

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1. Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

More information

Real Analysis Notes. Thomas Goller

Real Analysis Notes. Thomas Goller Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................

More information

In N we can do addition, but in order to do subtraction we need to extend N to the integers

In N we can do addition, but in order to do subtraction we need to extend N to the integers Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend

More information

Math LM (24543) Lectures 01

Math LM (24543) Lectures 01 Math 32300 LM (24543) Lectures 01 Ethan Akin Office: NAC 6/287 Phone: 650-5136 Email: ethanakin@earthlink.net Spring, 2018 Contents Introduction, Ross Chapter 1 and Appendix The Natural Numbers N and The

More information

Math 5210, Definitions and Theorems on Metric Spaces

Math 5210, Definitions and Theorems on Metric Spaces Math 5210, Definitions and Theorems on Metric Spaces Let (X, d) be a metric space. We will use the following definitions (see Rudin, chap 2, particularly 2.18) 1. Let p X and r R, r > 0, The ball of radius

More information

Set, functions and Euclidean space. Seungjin Han

Set, functions and Euclidean space. Seungjin Han Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

More information

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA  address: Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework

More information

Week 2: Sequences and Series

Week 2: Sequences and Series QF0: Quantitative Finance August 29, 207 Week 2: Sequences and Series Facilitator: Christopher Ting AY 207/208 Mathematicians have tried in vain to this day to discover some order in the sequence of prime

More information

1 Lecture 4: Set topology on metric spaces, 8/17/2012

1 Lecture 4: Set topology on metric spaces, 8/17/2012 Summer Jump-Start Program for Analysis, 01 Song-Ying Li 1 Lecture : Set topology on metric spaces, 8/17/01 Definition 1.1. Let (X, d) be a metric space; E is a subset of X. Then: (i) x E is an interior

More information

Continuity. Matt Rosenzweig

Continuity. Matt Rosenzweig Continuity Matt Rosenzweig Contents 1 Continuity 1 1.1 Rudin Chapter 4 Exercises........................................ 1 1.1.1 Exercise 1............................................. 1 1.1.2 Exercise

More information

Definition 2.1. A metric (or distance function) defined on a non-empty set X is a function d: X X R that satisfies: For all x, y, and z in X :

Definition 2.1. A metric (or distance function) defined on a non-empty set X is a function d: X X R that satisfies: For all x, y, and z in X : MATH 337 Metric Spaces Dr. Neal, WKU Let X be a non-empty set. The elements of X shall be called points. We shall define the general means of determining the distance between two points. Throughout we

More information

2 Topology of a Metric Space

2 Topology of a Metric Space 2 Topology of a Metric Space The real number system has two types of properties. The first type are algebraic properties, dealing with addition, multiplication and so on. The other type, called topological

More information

g 2 (x) (1/3)M 1 = (1/3)(2/3)M.

g 2 (x) (1/3)M 1 = (1/3)(2/3)M. COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is

More information

REAL VARIABLES: PROBLEM SET 1. = x limsup E k

REAL VARIABLES: PROBLEM SET 1. = x limsup E k REAL VARIABLES: PROBLEM SET 1 BEN ELDER 1. Problem 1.1a First let s prove that limsup E k consists of those points which belong to infinitely many E k. From equation 1.1: limsup E k = E k For limsup E

More information

Lecture 5 - Hausdorff and Gromov-Hausdorff Distance

Lecture 5 - Hausdorff and Gromov-Hausdorff Distance Lecture 5 - Hausdorff and Gromov-Hausdorff Distance August 1, 2011 1 Definition and Basic Properties Given a metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. If A is

More information

Immerse Metric Space Homework

Immerse Metric Space Homework Immerse Metric Space Homework (Exercises -2). In R n, define d(x, y) = x y +... + x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps

More information

REAL AND COMPLEX ANALYSIS

REAL AND COMPLEX ANALYSIS REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any

More information

THEOREMS, ETC., FOR MATH 515

THEOREMS, ETC., FOR MATH 515 THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every

More information

7 Complete metric spaces and function spaces

7 Complete metric spaces and function spaces 7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m

More information

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

More information

MAS221 Analysis, Semester 1,

MAS221 Analysis, Semester 1, MAS221 Analysis, Semester 1, 2018-19 Sarah Whitehouse Contents About these notes 2 1 Numbers, inequalities, bounds and completeness 2 1.1 What is analysis?.......................... 2 1.2 Irrational numbers.........................

More information

Week 5 Lectures 13-15

Week 5 Lectures 13-15 Week 5 Lectures 13-15 Lecture 13 Definition 29 Let Y be a subset X. A subset A Y is open in Y if there exists an open set U in X such that A = U Y. It is not difficult to show that the collection of all

More information

MATH 202B - Problem Set 5

MATH 202B - Problem Set 5 MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there

More information

ECARES Université Libre de Bruxelles MATH CAMP Basic Topology

ECARES Université Libre de Bruxelles MATH CAMP Basic Topology ECARES Université Libre de Bruxelles MATH CAMP 03 Basic Topology Marjorie Gassner Contents: - Subsets, Cartesian products, de Morgan laws - Ordered sets, bounds, supremum, infimum - Functions, image, preimage,

More information

Maths 212: Homework Solutions

Maths 212: Homework Solutions Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

More information

In N we can do addition, but in order to do subtraction we need to extend N to the integers

In N we can do addition, but in order to do subtraction we need to extend N to the integers Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need

More information

MATHS 730 FC Lecture Notes March 5, Introduction

MATHS 730 FC Lecture Notes March 5, Introduction 1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists

More information

MATH 131A: REAL ANALYSIS (BIG IDEAS)

MATH 131A: REAL ANALYSIS (BIG IDEAS) MATH 131A: REAL ANALYSIS (BIG IDEAS) Theorem 1 (The Triangle Inequality). For all x, y R we have x + y x + y. Proposition 2 (The Archimedean property). For each x R there exists an n N such that n > x.

More information

Chapter 3 Continuous Functions

Chapter 3 Continuous Functions Continuity is a very important concept in analysis. The tool that we shall use to study continuity will be sequences. There are important results concerning the subsets of the real numbers and the continuity

More information

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...

Functional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability... Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................

More information

Structure of R. Chapter Algebraic and Order Properties of R

Structure of R. Chapter Algebraic and Order Properties of R Chapter Structure of R We will re-assemble calculus by first making assumptions about the real numbers. All subsequent results will be rigorously derived from these assumptions. Most of the assumptions

More information

ABSTRACT INTEGRATION CHAPTER ONE

ABSTRACT INTEGRATION CHAPTER ONE CHAPTER ONE ABSTRACT INTEGRATION Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Suggestions and errors are invited and can be mailed

More information

M17 MAT25-21 HOMEWORK 6

M17 MAT25-21 HOMEWORK 6 M17 MAT25-21 HOMEWORK 6 DUE 10:00AM WEDNESDAY SEPTEMBER 13TH 1. To Hand In Double Series. The exercises in this section will guide you to complete the proof of the following theorem: Theorem 1: Absolute

More information

MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005

MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005 MATH 23b, SPRING 2005 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm (part 1) Solutions March 21, 2005 1. True or False (22 points, 2 each) T or F Every set in R n is either open or closed

More information

Math 117: Infinite Sequences

Math 117: Infinite Sequences Math 7: Infinite Sequences John Douglas Moore November, 008 The three main theorems in the theory of infinite sequences are the Monotone Convergence Theorem, the Cauchy Sequence Theorem and the Subsequence

More information

II - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define

II - REAL ANALYSIS. This property gives us a way to extend the notion of content to finite unions of rectangles: we define 1 Measures 1.1 Jordan content in R N II - REAL ANALYSIS Let I be an interval in R. Then its 1-content is defined as c 1 (I) := b a if I is bounded with endpoints a, b. If I is unbounded, we define c 1

More information

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski

Topology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology

More information

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011

Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure

More information

Part III. 10 Topological Space Basics. Topological Spaces

Part III. 10 Topological Space Basics. Topological Spaces Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.

More information

Introduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION

Introduction to Proofs in Analysis. updated December 5, By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Introduction to Proofs in Analysis updated December 5, 2016 By Edoh Y. Amiran Following the outline of notes by Donald Chalice INTRODUCTION Purpose. These notes intend to introduce four main notions from

More information

Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012

Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Definitions (2 points each) 1. State the definition of a metric space. A metric space (X, d) is set

More information

Chapter 8. P-adic numbers. 8.1 Absolute values

Chapter 8. P-adic numbers. 8.1 Absolute values Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.

More information

Lecture Notes on Metric Spaces

Lecture Notes on Metric Spaces Lecture Notes on Metric Spaces Math 117: Summer 2007 John Douglas Moore Our goal of these notes is to explain a few facts regarding metric spaces not included in the first few chapters of the text [1],

More information

Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010

Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010 Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

Economics 204 Summer/Fall 2011 Lecture 2 Tuesday July 26, 2011 N Now, on the main diagonal, change all the 0s to 1s and vice versa:

Economics 204 Summer/Fall 2011 Lecture 2 Tuesday July 26, 2011 N Now, on the main diagonal, change all the 0s to 1s and vice versa: Economics 04 Summer/Fall 011 Lecture Tuesday July 6, 011 Section 1.4. Cardinality (cont.) Theorem 1 (Cantor) N, the set of all subsets of N, is not countable. Proof: Suppose N is countable. Then there

More information

Based on the Appendix to B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University press,

Based on the Appendix to B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University press, NOTE ON ABSTRACT RIEMANN INTEGRAL Based on the Appendix to B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University press, 2003. a. Definitions. 1. Metric spaces DEFINITION 1.1. If

More information

3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?

3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure? MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable

More information

MATH NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS

MATH NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS MATH. 4433. NEW HOMEWORK AND SOLUTIONS TO PREVIOUS HOMEWORKS AND EXAMS TOMASZ PRZEBINDA. Final project, due 0:00 am, /0/208 via e-mail.. State the Fundamental Theorem of Algebra. Recall that a subset K

More information

Real Analysis. July 10, These notes are intended for use in the warm-up camp for incoming Berkeley Statistics

Real Analysis. July 10, These notes are intended for use in the warm-up camp for incoming Berkeley Statistics Real Analysis July 10, 2006 1 Introduction These notes are intended for use in the warm-up camp for incoming Berkeley Statistics graduate students. Welcome to Cal! The real analysis review presented here

More information

Math 117: Continuity of Functions

Math 117: Continuity of Functions Math 117: Continuity of Functions John Douglas Moore November 21, 2008 We finally get to the topic of ɛ δ proofs, which in some sense is the goal of the course. It may appear somewhat laborious to use

More information

Mathematics for Economists

Mathematics for Economists Mathematics for Economists Victor Filipe Sao Paulo School of Economics FGV Metric Spaces: Basic Definitions Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb. 2017 1 / 34 Definitions and Examples

More information

2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?

2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure? MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is

More information

1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty.

1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. 1. Supremum and Infimum Remark: In this sections, all the subsets of R are assumed to be nonempty. Let E be a subset of R. We say that E is bounded above if there exists a real number U such that x U for

More information

1. For each statement, either state that it is True or else Give a Counterexample: (a) If a < b and c < d then a c < b d.

1. For each statement, either state that it is True or else Give a Counterexample: (a) If a < b and c < d then a c < b d. Name: Instructions. Show all work in the space provided. Indicate clearly if you continue on the back side, and write your name at the top of the scratch sheet if you will turn it in for grading. No books

More information

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

More information

Summary of Real Analysis by Royden

Summary of Real Analysis by Royden Summary of Real Analysis by Royden Dan Hathaway May 2010 This document is a summary of the theorems and definitions and theorems from Part 1 of the book Real Analysis by Royden. In some areas, such as

More information

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE. B.Sc. MATHEMATICS V SEMESTER. (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION CORE COURSE B.Sc. MATHEMATICS V SEMESTER (2011 Admission onwards) BASIC MATHEMATICAL ANALYSIS QUESTION BANK 1. Find the number of elements in the power

More information

From now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1.

From now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1. Chapter 1 Metric spaces 1.1 Metric and convergence We will begin with some basic concepts. Definition 1.1. (Metric space) Metric space is a set X, with a metric satisfying: 1. d(x, y) 0, d(x, y) = 0 x

More information

Existence of a Limit on a Dense Set, and. Construction of Continuous Functions on Special Sets

Existence of a Limit on a Dense Set, and. Construction of Continuous Functions on Special Sets Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets REU 2012 Recap: Definitions Definition Given a real-valued function f, the limit of f exists at a point c R

More information

Analysis Qualifying Exam

Analysis Qualifying Exam Analysis Qualifying Exam Spring 2017 Problem 1: Let f be differentiable on R. Suppose that there exists M > 0 such that f(k) M for each integer k, and f (x) M for all x R. Show that f is bounded, i.e.,

More information

converges as well if x < 1. 1 x n x n 1 1 = 2 a nx n

converges as well if x < 1. 1 x n x n 1 1 = 2 a nx n Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists

More information

SOME QUESTIONS FOR MATH 766, SPRING Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm

SOME QUESTIONS FOR MATH 766, SPRING Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm SOME QUESTIONS FOR MATH 766, SPRING 2016 SHUANGLIN SHAO Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm f C = sup f(x). 0 x 1 Prove that C([0, 1]) is a

More information

Solve EACH of the exercises 1-3

Solve EACH of the exercises 1-3 Topology Ph.D. Entrance Exam, August 2011 Write a solution of each exercise on a separate page. Solve EACH of the exercises 1-3 Ex. 1. Let X and Y be Hausdorff topological spaces and let f: X Y be continuous.

More information

Chapter II. Metric Spaces and the Topology of C

Chapter II. Metric Spaces and the Topology of C II.1. Definitions and Examples of Metric Spaces 1 Chapter II. Metric Spaces and the Topology of C Note. In this chapter we study, in a general setting, a space (really, just a set) in which we can measure

More information

NOTES ON DIOPHANTINE APPROXIMATION

NOTES ON DIOPHANTINE APPROXIMATION NOTES ON DIOPHANTINE APPROXIMATION Jan-Hendrik Evertse January 29, 200 9 p-adic Numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics

More information

Econ Lecture 2. Outline

Econ Lecture 2. Outline Econ 204 2010 Lecture 2 Outline 1. Cardinality (cont.) 2. Algebraic Structures: Fields and Vector Spaces 3. Axioms for R 4. Sup, Inf, and the Supremum Property 5. Intermediate Value Theorem 1 Cardinality

More information