REVIEW OF ESSENTIAL MATH 346 TOPICS

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 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

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,.... 3 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 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 (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 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 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 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 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 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.