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 bounds of Then exists in and Theorem 1.20: The Archimedean property of Given, and There is a positive integer Theorem 1.20: such that is dense in If, and, then there exists a s.t. We can always find a rational number between two real numbers Theorem 1.21: -th Root of Real Numbers For every real, and positive integer There is one and only one positive real number Theorem 1.31: Properties of Complex Numbers If and are complex numbers, then is real and positive (except when ) Theorem 1.33: Properties of Complex Numbers If and are complex numbers, then Suppose unless in which case (Triangle Inequality) Theorem 1.37: Properties of Euclidean Spaces, then s.t.
Page 2 if and only if (Schwarz's Inequality) (Triangle Inequality) (Triangle Inequality) Theorem 2.8: Infinite Subset of Countable Set Every infinite subset of a countable set is countable Theorem 2.12: Union of Countable Sets be a sequence of countable sets, then Theorem 2.13: Cartesian Product of Countable Sets be a countable set be the set of all -tuples where Then for may not be distinct is countable Theorem 2.14: Cantor's Diagonalization Argument be the set of all sequqnecse whose digits are 0 and 1 Then is uncountable Theorem 2.19: Every Neighborhood is an Open Set Every neighborhood is an open set Theorem 2.20: Property of Limit Point If is a limit point of Then every neighborhood of Theorem 2.22: De Morgan's Law contains infinitely many points of be a finite or infinite collection of sets, then Theorem 2.23: Complement of Open/Closed Set A set is open if and only if is closed Note: This does not say that open is not closed and closed is not open
Page 3 Theorem 2.24: Intersection and Union of Open/Closed Sets Theorem 2.27: Properties of Closure If is a metric space and, then is closed is closed for every closed set s.t. Theorem 2.28: Closure and Least Upper Bound Property of If, and is bouned above, then Hence if is closed Theorem 2.34: Compact Sets are Closed Compact subsets of metric spaces are closed Theorem 2.35: Closed Subsets of Compact Sets are Compact Closed subsets of compact sets are compact Theorem 2.36: Cantor's Intersection Theorem If is a collection of compact subsets of a metric space s.t. The intersection of every finite subcollection of Theorem 2.37: Infinite Subset of Compact Set If Then is an infinite subset of a compact set has a limit point in Theorem 2.38: Nested Intervals Theorem If is a sequence of closed intervals in s.t. is nonempty
Page 4 Theorem 2.39: Nested -cell If be a positive integer is a sequence of -cells s.t. Theorem 2.40: Compactness of -cell Every -cell is compact Theorem 2.41: The Heine-Borel Theorem For a set is closed and bounded is compact Every infinite subset of, the following properties are equivalent has a limit point in Theorem 2.42: The Weierstrass Theorem Every bounded infinite subset of has a limit point in Theorem 2.47: Connected Subset of is connected if and only if If and, then be a sequence in a metric space has the following property Theorem 3.2: Important Properties of Convergent Sequences any neighborhood of contains for all but finitely many Given and. If converges to and to, then If converges, then is bounded Theorem 3.3: Algebraic Limit Theorem Theorem 3.4: Convergence of Sequence in Suppose where, then
Page 5 Theorem 3.6: Properties of Subsequence If is a sequence in a compact metric space Then some subsequence of Every bounded sequences in converges to a point of Theorem 3.10: Diameter and Closure contains a convergent subsequence If is the closure of a set in a metric space, then Theorem 3.10: Nested Compact Set If is a sequence of compact sets in s.t. Theorem 3.11: Cauchy Sequence and Convergence In any metric space, every convergent sequence is a Cauchy sequence If is a compact metric space and is a Cauchy sequence Then In converges to some point of, every Cauchy sequence converges Theorem 3.14: Monotone Convergence Theorem If is monotonic, then converges if and only if it is bounded Theorem 3.17: Properties of Upper Limits If Moreover be a sequence of real numbers, then is the only number with these properties Theorem 3.20: Some Special Sequences Theorem 3.22: Cauchy Criterion for Series
Page 6 Theorem 3.23: Series and Limit of Sequence Theorem 3.24: Convergence of Monotone Series A series of nonnegative real numbers converges if and only if its partial sum form a bounded sequence Theorem 3.25: Comparison Test Theorem 3.26: Convergence of Geometric Series If Suppose > 1, the series diverges Theorem 3.27: Cauchy Condensation Test, then Theorem 3.28: Convergence of Series Theorem 3.33: Root Test
Page 7 Theorem 3.34: Ratio Test Theorem 3.39: Convergence of Power Series Theorem 3.43: Alternating Series Test Suppose we have a real sequence s.t. Theorem 3.45: Property of Absolute Convergence If converges absolutely, then converges Theorem 3.54: Riemann Series Theorem be a series of real number which converges nonabsolutely Then there exists a rearrangement Theorem 3.55: Rearrangement and Absolute Convergence If is a series of complex numbers which converges absolutely Then every rearrangement of Suppose be a metric space, and be a limit point of s.t. converges to the same sum Theorem 4.4: Algebraic Limit Theorem of Functions
Page 8 be complex functions on where Then Theorem 4.6: Continuity and Limits In the context of Definition 4.5, if is also a limit point of, then Theorem 4.7: Composition of Continuous Function Suppose are metric spaces,, and defined by If is continuous at, and is continuous at Then is continuous at Theorem 4.8: Characterization of Continuity Given metric spaces Statement is continuous if and only if is open in for every open set Theorem 4.14: Continuous Functions Preserve Compactness be metric spaces, compact If is continuous, then is also compact Theorem 4.15: Applying Theorem 4.14 to be a compact metric space If is continuous, then is closed and bounded Thus, is bounded Theorem 4.16: Extreme Value Theorem be a continuous real function on a compact metric space Then s.t. and Equivalently, s.t.
Page 9 Theorem 4.17: Inverse of Continuous Bijection is Continuous be metric spaces, compact Suppose Define Then is continuous and bijictive by is also continuous and bijective Theorem 4.19: Uniform Continuity and Compactness be metric spaces, compact If is continuous, then is also uniformly continuous Theorem 4.20: Continuous Mapping from Noncompact Set be noncompact set in Then there exists a continuous function on s.t. is not bounded is bounded but has no maximum is bounded, but be metric spaces be a continuous mapping is not uniformly continuous Theorem 4.22: Continuous Mapping of Connected Set If is connected then is also connected Theorem 4.23: Intermediate Value Theorem be continuous on If and if statifies Then s.t. Theorem 5.2: Differentiability Implies Continuity be defined on If is differentiable at then is continuous at Theorem 5.5: Chain Rule Given If If is continuous on, and exists at is defined on, and is differentiable at, then is differentiable at, and Theorem 5.8: Local Extrema and Derivative be defined on has a local maximum (or minimum) at
Page 10 Then Given and if it exists Theorem 5.9: Extended Mean Value Theorem are continuous real-valued functions on are differentiable on Then there is a point at which Theorem 5.10: Mean Value Theorem If is continuous on and differentiable on Then Suppose s.t. Theorem 5.11: Derivative and Monotonicity is differentiable on If, then is monotonically increasing If, then is constant If, then is monotonically decreasing Theorem 5.15: Taylor's Theorem Suppose is a real-valued function on Fix a positive integer is continuous on exists, where Then between and s.t. Theorem 6.4: Properties of Refinement If is a refinement of, then Theorem 6.5: Properties of Common Refinement
Page 11 Theorem 6.6 Theorem 6.8 on if and only if there exists a partition s.t. If is continuous on, then on Theorem 6.9 If is monotonic on, and is continuous on Then Theorem 6.10 on If is bounded on with finitely many points of discontiunity And is continuous on these points, then Theorem 6.20: Fundamental Theorem of Calculus (Part I) on is continuous on Furthermore, if is continuous at, then is differentiable at on, and Theorem 6.21: Fundamental Theorem of Calculus (Part II) If there exists a differentiable function on s.t.
Page 12 Number Systems, Irrationality of Wednesday, January 24, 2018 12:01 PM Course Overview The real number system Metric spaces and basic topology Sequences and series Continuity Topics from differential and integral calculus Grading Homework assignments 20% Quiz (Feb. 9) 5% Midterm 1 (Mar. 9) 20% Midterm 2 (Apr. 13) 20% Final (May 10 @ 7:45-9:45 AM) 35% A 9 % B 8 % C 7 % D 6 % Tutoring Tom Stone @VV B205 Monday 2:30-4:30 PM Tuesday 2:00-4:00 PM Number Systems Natural Numbers: Integers: Rational Numbers: Real numbers : fill the "holes" in the rational numbers Example 1.1: Irrationality of There is no rational number such that Proof by contradiction Assume there is a rational number p such that
Page 13 So is even So is also even are both division by 2 This contradicts the fact that So no such exists have no common factor
Page 14 Sets, Gaps in Q, Field Friday, January 26, 2018 12:03 PM Definition 1.3: Sets Contains Set Subset If is a set and is an element of, then we write Otherwise, we write The empty set or null set is a set with no elements, and is denoted as If a set has at least one element, it is called nonempty If and are sets and every element of is an element of Then is a subset of Rubin write this Fact: Proper subset Equal If for all sets contain something not in A, then A is a proper subset of B If and then Otherwise Example 1.1: Gaps in Rational Number System We have proved that i.e. there is no rational number is not rational such that Prove: A has no largest element, and B has no smallest element If
Page 15 If i.e. A has no largest element A field is a set i.e. B has no smallest element Definition 1.12: Field with two binary operations called addition and multiplication that satisfy that following field axioms Example Axioms for addition (+) (A1) If and, then (A2) Addition is communicate: (A3) Addition is associative: (A4) There exists s.t. (A5), there exists an additive inverse s.t. Axioms for multiplication ( ) (M1) If and, then (M2) Addition is communicate: (M3) Addition is associative: (M4) contains an element s.t. (M5) If, then there exists s.t. (D) The distributive law: The real numbers are an example of field
Page 16 Field, Order, Ordered Set Monday, January 29, 2018 12:00 PM Proposition 1.14: Properties of Fields (Addition) Given a field, for (1) (2) (3) (4) If If If, then, then, then by (A5) by (A3) by (A2) by (A6) by (A4) (1) (2) Proposition 1.15: Properties of Fields (Multiplication) Given a field, for If and, then If and, then (3) If and, then (4) If, then Proof similar to Proposition 1.14 Proposition 1.16: Properties of Fields Given a field, for (1)
Page 17 (2) If and, then Suppose, but, so exists (3) (4) This is a contradiction, so And the rest is similar Intuition Use (3), Definition 1.5: Order The real number line Definition be a set. An order on is a relation, denoted by with the following two properties: Other notations Definition If, then only one of the statements is true If, if and, then (Transitivity) means either means either Definition 1.6: Ordered Set or or
Page 18 An ordered set is a set for which an order is defined. Example is an ordered set under the definition that for if and only if is positive
Page 19 Infimum and Supremum, Ordered Field Wednesday, January 31, 2018 12:00 PM Definition 1.7: Upper Bound and Lower Bound Suppose If there exists is an ordered set and such that We say that is bounded above and call an upper bound for If there exists such that We say that is bonded below by, and is a lower bound for Definition 1.8: Least Upper Bound and Greatest Lower Bound Definition Recall Suppose is an ordered set and is bounded above. Suppose there exists is an upper bond of s.t. If, then is not an upper bound of Then we call Suppose there exists is an lower bond of the least upper bound (or lub or sup or supremium) of s.t. If, then is not an lower bound of Then we call the greastst lower bound (or glb or inf or infimum) of Examples 1.9: Least Upper Bound and Greatest Lower Bound doesn't exist doesn't exist has no sup in has no inf in If exists, may or may not be in
Page 20 Definition 1.10: Least-Upper-Bound property We say that a ordered set has least-upper-bound property provided that if s.t. and is bounded above, then exists and Theorem 1.11: Greatest-Lower-Bound Property Statement Proof Suppose is an ordered set with the least-upper-bound property Suppose, and is bounded below be the set of lower bounds of Then exists in and is bounded below, so is bounded above is not empty Given, we have by definition of Therefore, And So exists in is bounded above is bounded above has least upper bound property exists is a lower bound for (i.e. ) If, then is not an upper bound for, so So Thus, i.e. If for all is a lower bound for is another lower bound for since So,, but if Therefore i.e. Therefore Definition 1.17: Ordered Field Definition An ordered field is a field is an upper bound for is the least upper bound of if which is also an ordered set, such that and
Page 21 Examples Note if, and If, we call positive If, we call negative is an ordered field with least-upper-bound property
Page 22 Ordered Field, Archimedean Property, Friday, February 2, 2018 12:05 PM is dense in Proposition 1.18: Properties of Ordered Field (1) (2) (3) (4) be an ordered field, for If then, and vice versa If and then If and then By (1), By (2), By (1), If then. In particular If, by (2), If, by (3), So (5)
Page 23 Theorem 1.19: Least-Upper-Bound Property of There exists an ordered filed with the least-upper-bond property called Moreover has as a subfield Proof: See appendix Theorem 1.20: The Archimedean property of Statement Proof Given, and There is a positive integer such that Assume the Archimedean property is false Then i.e. And, so has an upper bound exists By definition of So, Corollary This contradicts is not an upper bound for for some positive integer Therefore the Archimedean property is true Given Theorem 1.20: Statement, then s.t. is dense in If, and, then there exists a s.t. We can always find a rational number between two real numbers
Page 24 Proof So, and By the Archimedean property of There exists a positive integer By the Archimedean property of There are positive integers i.e. So there is an integer And more importantly, s.t. s.t. again Combining two parts together, we have s.t.
Page 25 n-th Root of Real Numbers Monday, February 5, 2018 12:10 PM Theorem 1.21: -th Root of Real Numbers Notation For a positive integer For a negative integer Statement For every real, and positive integer There is one and only one positive real number s.t. Intuition Try this for and, so Proof (Uniqueness) Lemma If there were and s.t., but Without loss of generality, assume Then, so they can't both equal So, there is at most one such If is a positive integer, then Moreover, if, then Proof (Existence) is not empty So,
Page 26 Thus, Therefore is bounded above Therefore is not empty So and is bounded above by By least upper bound property, We now show that Assume and exists Then Use the lemma Set Since is not an upper bound of This contradicts Thus, Assume, then Use the lemma Set Therefore, By definition of
Page 27 and is greater than everything in Also, so is an upper bound for But Thus, Therefore, which contradicts Corollary: If, and then So, then
Page 28 Complex Numbers, Euclidean Spaces Wednesday, February 7, 2018 12:12 PM Complex Numbers Definition If, then where and Addition, multiplication, subtraction, and division If For, then Real part and imaginary part Complex conjugate Absolute value is the real part of is the imaginary part of is the complex conjugate of is the absolute value of Note For a real number Complex division If, then Theorem 1.31: Properties of Complex Numbers If and are complex numbers, then
Page 29 (1) (2) is real and positive (except when ) Theorem 1.33: Properties of Complex Numbers If and are complex numbers, then (3) unless Then in which case (4) (5) So Thus, (Triangle Inequality) by (4) by (3) by (2) Definition 1.36: Euclidean Spaces
Page 30 Inner product If with Then the inner product of and is Norm If, we define the norm of to be Euclidean spaces Suppose The vector space if and only if with inner product and norm is called Euclidean -space Theorem 1.37: Properties of Euclidean Spaces Statement, then (Schwarz's Inequality) (Triangle Inequality) (Triangle Inequality) Theorem 1.35: Schwarz Inequality Proof See Theorem 1.35 in Rudin for a proof of Schwarz Inequality for For intuition, try proving Triangle Inequality In a Euclidean Space, Thus, we have
Page 31 Function, Cardinality, Equivalence Relation Monday, February 12, 2018 12:08 PM Definition 2.1 & 2.2: Function Given two sets and A function (or mapping) is a rule that assigns elements in Notationally, if is a function from to, we write to elements in Set Set is called the domain of is called the codomain of For, is the image of under is called the range of If, then we say that is onto or surjective If, then is one-to-one or injective A function that is both one-to-one and onto is said to be bijective For, is the inverse image of under
Page 32 Notationally, if, Example is at most a single element set for all if and only if is injective In this case, defined by Definition 2.3: Cardinality can be thought of as a function maps to the single element, we can also write If there exists a one-to-one, onto mapping from set to set We say that and can be put in one-to-one correspondence And that and have the same cardinality (or cardinal number) In this case, we write Definition 2.3: Equivalence Relation One-to-one correspondence is an example of an equivalence relation An equivalence relation satisfies 3 properties Reflexive: Symmetric: If Transitivity: If, then, then
Page 33 Cardinality and Countability, Sequence Wednesday, February 14, 2018 12:06 PM Definition 2.4: Cardinality and Countability For any set, we say and is finite if for some ( is also considered as finite) is infinite if is countable if is uncountable if is at most coutable if is countable Define for all Examples 2.5: Countability is countable is neither finite nor countable by is finite or countable is injective If Either way, is surjective Thus Given, If If is countable is bijective There are "less" rational numbers ( ) than there are ordered pairs of integers We can also ignore negatives and zeros
Page 34 because integers are in 1-1 correspondence with Idea: Write ordered pairs of integers in a 2 dimension array Putting this all together, we have Definition 2.7: Sequence Definition Example A sequence is a function defined on Notationally, this is often written Meaning Statement Intuition for all Theorem 2.8: Infinite Subset of Countable Set Proof Every infinite subset of a countable set is countable Countable sets represent the "smallest" infinity No uncountable set can be a subset of a countable set. Suppose is countable and is infinite
Page 35 Since is countable, its element will be a sequqnce (order given by the bijective function ) be the smallest such that be the next smallest such that So i.e. is a sequence indexed by Now consider given by is clearly one-to-one and onto by construction Therefore is countable Example 9 6 We can show that is a bijection Thus, is countable
Page 36 Set Operations, Countable and Uncountable Friday, February 16, 2018 12:08 PM Definition 2.9: Set-Theoretic Operations Set theoretic union Set theoretic intersection Indexing set Example Theorem 2.12: Union of Countable Sets Statement be a sequence of countable sets, then Proof Just like the proof that is countable Go along the diagonal, we have Corollary
Page 37 Suppose is at most countable If is at most countable Theorem 2.13: Cartesian Product of Countable Sets Statement Proof be a countable set be the set of all -tuples where Then for may not be distinct is countable We proof by induction on Base case: Here, Now assume for are all the elements of A with possible repetition The set of -tuples are countable Now we treat the where as ordered pairs Statement By case, the set of is still countable Theorem 2.14: Cantor's Diagonalization Argument be the set of all sequqnecse whose digits are 0 and 1 Then Suppose is uncountable Proof: Cantor's Diagonalization Argument is countable Then where is a sequence of and for all
Page 38 where for Construct a new sequence where Then So, which is a contradiction Thus, must be uncountable Corollary is uncountable
Page 39 Metric Space, Interval, Cell, Ball, Convex Monday, February 19, 2018 12:04 PM Definition 2.15: Metric Space Definition A set of points is called a metric space if there exists a metric or distance function such that Positivity if and Example 1 Symmetry for all Triangle Inequality for all for all If, this is just standard numerical absolute value and is the distance on the number line Example 2 (Taxicab metric) where Is this a true metric space? Positivity Clearly since it is a sum of absolute values
Page 40 Suppose i.e. Suppose Thus Symmetry Triangular Inequality by Triangle Inequality of Interval Definition 2.17: Interval, -cell, Ball, Convex -cell Ball Segment is (open interval) Interval is (closed interval) We can also have half-open intervals: If for The set of points in and that satisfy ( ) is called a -cell If and The open ball with center with radius is
Page 41 the closed ball with center with radius is Convex We call a set convex if i.e. All points along a straight line from to and between and is in Example: Balls are convex Given an open ball with center If, then and Thus i.e. and radius by Triangle Inequality
Page 42 Definitions in Metric Space Wednesday, February 21, 2018 12:01 PM Definitions 2.18: Definitions in Metric Space be a metric space. All points/elements below are in Neighborhood Definition A neighborhood of is a set consisting of all points such that for some We call Example: the radius of Example: Taxicab metric Limit point Definition A point is a limit point of the set if every neighborhood of contains a point and Example:
Page 43 Example: For Isolated point Definition, the limit points is If and is not a limit point of, then is an isolated point of Example: in Every integers is an isolated point in Closed set Definition A set is closed if every limit point of is in Example: In, neighborhood of are open intevals cenerted about All of If is a limit point since The neighborhood about is is non-empty If, then take If Otherwise take So every point in i.e. or is a limit point
Page 44 Take So Thus Interior point Definition Then So nothing outside of contains all its limit points is closed is a limit point of A point is an interior point of a set if there exists a neighborhood Example: For the closed set The point is an interior point of that is a subset of The point is not an interior point of (on the boundary of ) Open set Definition is an open set if every point of Example: is an open set, since is an interior point
Page 45 Example: For Take Complement Perfect Bounded Dense Thus every point in is an interior point The complement of (denoted as ) is is perfect if is closed and every point of is limit point of is bounded if there is a real number and a point s.t. is dense in for all if every point of is a limit point of or a point of (or both)
Page 46 Neighborhood, Open and Closed, De Morgan's Law Friday, February 23, 2018 12:06 PM Theorem 2.19: Every Neighborhood is an Open Set Statement Proof Every neighborhood is an open set be a metric space Choose a neighborhood Choose s.t. Consider the neighborhood Thus i.e. So Therefore is open Theorem 2.20: Property of Limit Point Statement
Page 47 If is a limit point of Then every neighborhood of contains infinitely many points of Proof Suppose the opposite Then there exists a set with a limit point s.t. The neighborhood of contains only finitely many points of Namely By definition, for This contradicts the fact that is a limit point So, this neighborhood about must contain infinitely many points Corollary A finite set has no limit points Theorem 2.22: De Morgan's Law Statement be a finite or infinite collection of sets, then Proof
Page 48 So Thus, for all Proof Then So for all for all Theorem 2.23: Complement of Open/Closed Set Statement Proof A set is open if and only if is closed Note: This does not say that open is not closed and closed is not open Suppose Choose So, is closed, so is not a limit point of i.e. There exists a neighborhood So, Consequently, So, is an interior point of By definition, is open that contains no points of
Page 49 Proof Suppose is open be a limit point of (if exists) So, every neighborhood of So, is not an interior point of is open, so Thus, Corollary contains a point in contains its limit points and is closed by definition A set is closed if and only if is open Examples 2.21: Closed, Open, Perfect and Bounded Subset Closed Open Perfect Bounded A nonempty finite set? Note: is open as a subset of, but not as a subtset of
Page 50 Open and Closed, Closure Monday, February 26, 2018 12:06 PM Theorem 2.24: Intersection and Union of Open/Closed Sets (a) Suppose is open for all If, then for some Since is open, there is a neighborhood about in And consequently, the neighborhood about is also in Thus is open (b) Suppose is closed for all Then is open by Theorem 2.23 (c) Suppose is open So, for By definition, since each is open is contained in a neighborhood
Page 51 So, for (d) Suppose is closed Then is open by Theorem 2.23 Note Definition 2.26: Closure be a metric space If and denotes the set of limit points of in Then the closure of is defined to be Theorem 2.27: Properties of Closure If is a metric space and, then is closed Then is neither a point of nor a limit point of So there exists a neighborhood about that contains no points of i.e. every point of Thus is open is an interior point
Page 52 Therefore is closed is closed If, then is closed If is closed, contains its limit points, so and for every closed set Suppose Thus is closed and Intuition: is the smallest closed set in containing Statement s.t. Theorem 2.28: Closure and Least Upper Bound Property of Proof If, and is bouned above, then Hence if is closed If If Clearly Suppose, then is an upper bound for But this contradicts the fact that So there must be some with Thus, for any neighborhood about, So i.e. is a limit point of in the neighborhood
Page 53 Convergence and Divergence, Range, Boundedness Wednesday, February 28, 2018 12:07 PM Definition 3.1: Convergence and Divergence Definition A sequence in a metric space converges to a point if Given any, s.t. If converges to, we write If does not converge, it is said to diverge Intuition is small is a "point of no return" beyond which sequence is within of Range Given a sequence The set of points ( ) is called the range of the sequence Range could be infinite, but it is always at most countable Since we can always construct a function Boundedness A sequence, where is said to be bounded if its range is bounded Examples of Limit, Range and Boundedness Consider the following sequences of complex numbers
Page 54 Limit Range Bounded 0 Infinite Yes Divergent Infinite No 1 Infinite Yes Divergent Yes 1 Yes
Page 55 Important Properties of Convergent Sequences Friday, March 2, 2018 12:06 PM Theorem 3.2: Important Properties of Convergent Sequences be a sequence in a metric space Suppose be a neighborhood of with radius So, any neighborhood of contains for all but finitely many converges to may not be in Suppose every neighborhood of be given, but there are only finitely many of these contains all but finitely many is a neighborhood of By assumption, all but finitely points in Choose Then s.t. are in Given and. If converges to and to, then be given, then Since is arbitrary, Therefore If converges, then is bounded Since converges to some, then s.t. Then By definition, is bounded
Page 56 Since is a limit point of Every neighborhood of contains, and be given Therefore
Page 57 Algebraic Limit Theorem Monday, March 5, 2018 12:10 PM Theorem 3.3: Algebraic Limit Theorem Given, then for Given So, Given If If Standard approach Rudin's approach Given
Page 58 s.t. for s.t. for for By the Triangle Inequality,
Page 59
Page 60 Sequence Convergence in Wednesday, March 7, 2018 12:15 PM, Compact Set Theorem 3.4: Convergence of Sequence in Statement (a) Proof (a) Suppose where, then Assume Given, there exists s.t. for Thus, for Statement (b) Therefore Suppose Then Proof (b) and are sequences in, is a sequence in This follows from (a) and Theorem 3.3 (Algebraic Limit Theorem) Definition 2.31: Open Cover An open cover of a set in a metric is Definition 2.32: Compact Sets
Page 61 Definition Intuition for A set in a metric space is compact if every open cover of : Closed and bounded has a finite subcover Example 1 is a open cover of itself, but is not compact We cannot take a finite collection of these So it has no finite subcover Therefore is not compact Example 2 Consider, and still have an open cover Then Therefore for some is an open cover of is compact
Page 62 Compact Subset, Cantor's Intersection Theorem Monday, March 12, 2018 12:08 PM Theorem 2.34: Compact Sets are Closed Statement Proof Compact subsets of metric spaces are closed be a compact subset of a metric space We shall prove that the complement of is open Since is compact, s.t. Then is a neighborhood of that does not intersect is an interior point of So is open and therefore is closed
Page 63 Theorem 2.35: Closed Subsets of Compact Sets are Compact Statement Proof Closed subsets of compact sets are compact be a metric space Suppose, where is closed, and is compact be an open cover of Consider, where is open Then is an open cover of Since is compact, has a finite subcover If, then is still finite and covers So we have a finite subcover of Therefore is compact
Page 64 Corollary Proof If is closed and is compact, then is compact compact is closed We know is closed, so is closed So Statement, and is compact is compact Theorem 2.36: Cantor's Intersection Theorem Proof If is a collection of compact subsets of a metric space s.t. The intersection of every finite subcollection of Fix and let Assume no point of Then Since Where Then is an open cover of is compact, belongs to every is a finite collection of indices is nonempty
Page 65 This is a contradiction, so no such set exists The result follows Corollary If is a sequence of nonempty compact sets s.t. Theorem 2.37: Infinite Subset of Compact Set Statement Proof If Then is an infinite subset of a compact set If no point of has a limit point in were a limit point of Then, s.t. no point of other than i.e. contains at most one point of E (namely,, if ) So no finite sub-collection of This is a contradiction, so can cover, and thus not has a limit point in
Page 66 Nested Intervals Theorem, Compactness of -cell Wednesday, March 14, 2018 12:06 PM Theorem 2.38: Nested Intervals Theorem Statement If is a sequence of closed intervals in s.t. Intuition Proof So, is nonempty is bounded above by So, exists since Theorem 2.39: Nested -cell Statement If be a positive integer is a sequence of -cells s.t.
Page 67 Proof consists of all points s.t. For each, satisfies the hypothesis of Theorem 2.38 Theorem 2.40: Compactness of -cell Statement Proof Every -cell is compact be a -cell Suppose Build sequence is an open cover of with no finite subcover Those intervals describes -cells whose union is Since the number of is finite, and has no finite subcover not covered by a finite subcover of Repeat this process on We can build a sequence is a sequence of -cells s.t. to obtain is not covered by any finite sub-collection of ; call this
Page 68 By Theorem 2.38, Then i.e. is open, for some s.t. In this case,, which is impossible, since is not covered by any finite sub-collection of So no such open cover exists So every open cover of have a finite subcover Therefore is compact
Page 69 Heine-Borel, Weierstrass, Subsequence Friday, March 16, 2018 12:07 PM Theorem 2.41: The Heine-Borel Theorem For a set (a) (b) (c) Proof Proof Proof is closed and bounded is compact, the following properties are equivalent Every infinite subset of If (a) holds, then (b) follow from has a limit point in for some -cell Theorem 2.40 ( is compact) Theorem 2.35 (Closed subsets of compact sets are compact) See Theorem 2.37 Suppose is not bounded s.t. is an infinite subset of This is a contradiction, so Suppose is not closed that is a limit point of with no limit points must be bounded but not in be a infinite subset of By construction, has as a limit point We want to show that and By triangle inequality, is the only limit point of There are only finitely many points of in it cannot be a limit point of Since was arbitrary, is the only limit point of
Page 70 Statement By (c), has a limit point in i.e. This is a contradiction, s Therefore is closed and bounded has to be closed Theorem 2.42: The Weierstrass Theorem Proof Every bounded infinite subset of has a limit point in is bounded, so By Theorem 2.40, is compact By Theorem 2.37, Hence, Given a sequence has a limit point in Definition 3.5: Subsequences Definition Consider a sequence for some has a limit point in with Then the sequence is a subsequence of If converges, its limit is called a subsequential limit of Example 6 7 8 Note Statement (a) 9 8 7 9 A subsequential limit might exist for a sequence in the absence of a limit If converges to if and only if every subsequence of converges to Theorem 3.6: Properties of Subsequence Proof (a) is a sequence in a compact metric space Then some subsequence of If be the range of is finite converges to a point of and a sequence with s.t.
Page 71 If is infinite By Theorem 2.37, It follows that has a limit point converges to Statement (b) Proof (b) Every bounded sequences in By Theorem 2.41, every bounded subset of Result follows by (a) contains a convergent subsequence is in a compact subset of
Page 72 Cauchy Sequence, Diameter Monday, March 19, 2018 12:19 PM Definition 3.8: Cauchy Sequence A sequence in a metric space is said to be Cauchy sequence If s.t., Definition 3.9: Diameter be a nonempty subset of metric space be set of all real numbers of the form with Then is called the diameter of (possibly ) If is a sequence in and Theorem 3.10: Diameter and Closure Statement Proof If is the closure of a set in a metric space, then This is obvious since
Page 73 Statement So Therefore Since was arbitrary, Theorem 3.10: Nested Compact Set If is a sequence of compact sets in s.t. Proof By Theorem 2.36, is not empty If contains more than one point, But, then There can only be one point in
Page 74 Cauchy Sequence, Complete Metric Space, Monotonic Wednesday, March 21, 2018 12:07 PM Theorem 3.11: Cauchy Sequence and Convergence Statement (a) Proof (a) In any metric space, every convergent sequence is a Cauchy sequence Suppose So Statement (b) is a Cauchy sequence If is a compact metric space and is a Cauchy sequence Then Proof (b) converges to some point of be a Cauchy sequenece in compact metric space For, let By Theorem 2.35, as closed subset of is compact Since By Theorem 3.10 (b), s.t. Since In other word, Statement (c) In Proof (c), every Cauchy sequence converges be a Cauchy sequence in Then the range of is By Theorem 2.41, every bounded subset of has compact closure in
Page 75 Definition Examples (c) follows from (b) Definition 3.12: Complete Metric Space A sequence A metric space is said to be complete if every Cauchy sequence converges in is complete Compact metric space is complete is not complete (convergence may lie outside of ) Definition 3.13: Monotonic Sequence monotonically increasing if monotonically decreasing if monotonic if Statement of real numbers is said to be is either monotonically increasing or decreasing Theorem 3.14: Monotone Convergence Theorem Proof If is monotonic, then converges if and only if it is bounded By Theorem 3.2 (c), converge implies boundedness Without loss of generality, suppose, and, then Given, is monotonically increasing Since is not an upper bound of, and is increasing
Page 76 Upper and Lower Limits Friday, March 23, 2018 12:11 PM Definition 3.15: Sequences Approaching Infinity Then we write Similarly if Then we write Definition be a sequence of real numbers s.t. s.t. s.t. Definition 3.16: Upper and Lower Limits be a sequence of real numbers be the set of (in the extended real number system) s.t. for some subsequence contains all subsequential limits of plus possibly Example 1 6 Example 2 All subsequential limits of a convergent sequence converge to the same value as the sequence All subsequential limits = Theorem 3.17: Properties of Upper Limits
Page 77 be a sequence of real numbers, then When is not bounded above, so is not bounded above There is a subseqnence s.t. If So is bounded above And at least one subsequential limit exists i.e. By Theorem 3.7, By Theorem 2.28, Therefore Then and is closed i.e. If with for infinitely many Then Moreover s.t. This contradicts the definition of Suppose Without loss of generality, suppose Choose Since is the only number with these properties s.t. satisfies the property above s.t. So no subsequence of This contradicts the existence of s.t. the property above holds for can converge to Therefore only one number can have these properties
Page 78 Some Special Sequences Monday, April 2, 2018 12:11 PM Theorem 3.20: Some Special Sequences Lemma (The Squeeze Theorem) Given where is some fixed number If, then (Proof on homework) When Then, then By the Squeeze Theorem, When
Page 79 s.t. by
Page 80 Series, Cauchy Criterion for Series, Comparison Test Wednesday, April 4, 2018 12:09 PM Definition 3.31: Series Given a sequence If diverges, the series is said to diverge is called the sum of the series But it is technically the limit of a sequence of sums Theorem 3.22: Cauchy Criterion for Series Statement Proof Statement This is Theorem 3.11 applied to Theorem 3.23: Series and Limit of Sequence In the setting of Theorem 3.22, take We have for Note 6 7 8 8 8 8 8
Page 81 Theorem 3.24: Convergence of Monotone Series Statement Proof A series of nonnegative real numbers converges if and only if its partial sum form a bounded sequence See Theorem 3.14 (Monotone Convergence Theorem) Theorem 3.25: Comparison Test Theorem 3.26: Convergence of Geometric Series Statement Note If > 1, the series diverges This only works if we know this series converges Proof If, we have
Page 82
Page 83 Convergence Tests for Series Friday, April 6, 2018 12:06 PM Theorem 3.27: Cauchy Condensation Test Statement Suppose, then Proof By Theorem 3.24, we just need to look at boundness of partial sums For For For Statement So and are both bounded or unbounded Theorem 3.28: Convergence of Series Proof If
Page 84 Cauchy Condensation Test, By Theorem 3.26, this converges if Otherwise, Theorem 3.33: Root Test, and this diverges Theorem 3.17(b) says if and s.t. By Theorem 3.17, there exists a sequence s.t. So for infinitely many, i.e. Theorem 3.34: Ratio Test Statement
Page 85 Proof In particular So, Note On the other hand, if Then, so series divreges by Theorem 3.23
Page 86 Power Series, Absolute Convergence, Rearrangement Monday, April 9, 2018 12:10 PM Definition 3.38: Power Series Given a sequence of complex numbers Theorem 3.39: Convergence of Power Series Statement Proof and apply the root test Note: Examples is called the radius of convergence of the power series Theorem 3.43: Alternating Series Test Statement
Page 87 Suppose we have a real sequence s.t. Proof: HW Example: alternating harmonic series Absolute Convergence If converges but diverges We way that Statement Proof converges nonabsolutely or conditionally Theorem 3.45: Property of Absolute Convergence If converges absolutely, then converges The result follows by Cauchy Criterion Definition 3.52: Rearrangement be a sequence in which every natural number appears exactly once, then is called a rearrangement of Theorem 3.54: Riemann Series Theorem be a series of real number which converges nonabsolutely Then there exists a rearrangement Theorem 3.55: Rearrangement and Absolute Convergence Statement If s.t. is a series of complex numbers which converges absolutely Then every rearrangement of converges to the same sum
Page 88 Proof be a rearrangement of with partial sum By the Cauchy Criterion, given s.t. Choose s.t. are all contained in the set Where Then if So, are the indices of the rearranged series will be cancelled in the difference converges to the same value as
Page 89 Limit of Functions Wednesday, April 11, 2018 12:15 PM Definition 4.1: Limit of Functions Definition Note be metric spaces, and Suppose and is a limit point of s.t. is the deleted neighborhood about and refer to the distances in and, respectively Relationship with sequence If of radius Theorem 4.2 relates this type of limit to the limit of a sequence Consequently, if Definition 4.3: Algebra of Functions Suppose be a metric space, and be a limit point of has a limit at, then its limit is unique, then we define Theorem 4.4: Algebraic Limit Theorem of Functions be complex functions on where Then
Page 90 Continuous Function and Open Set Monday, April 16, 2018 12:09 PM Definition 4.5: Continuous Function Definition Note Suppose are metric spaces,, and Then is continuous at if For every, there exists s.t. If is continuous at every point, then is continuous on must be defined at to be continous at (as opposed to limit) Every function is continuous at isolated point Theorem 4.6: Continuity and Limits In the context of Definition 4.5, if Statement Note is also a limit point of, then Theorem 4.7: Composition of Continuous Function Proof Suppose are metric spaces,, and defined by If is continuous at, and is continuous at Then is continuous at is called the composition of and and is written as be given Since is continuous at s.t. If and, then Since is continuous at, s.t. If and, then Consequently, if, and, then So, is continuous at by definition
Page 91 Theorem 4.8: Characterization of Continuity Statement Proof Given metric spaces is continuous if and only if is open in for every open set Suppose is continuous on, and is open We want to show that all points of Suppose, and, then Since is open There exists a neighborhood of are interior points that is a subset of In other word, s.t. Since is continuous at s.t. Suppose Proof Then, since is open Thus, This shows that Therefore is an interior point of is open in Suppose is open in for every open set and fix be the neighborhood of Since is open, is also open by assumption Thus, s.t. But if, then, and so So, Corollary Proof is continuous at Since was arbitrary, is continuous on Given metric spaces is continuous on if and only if is closed in for every closed set in
Page 92 A set is closed if and only if its complement is open Also,, for every
Page 93 Continuity and Compactness, Extreme Value Theorem Wednesday, April 18, 2018 12:06 PM Definition 4.13: Boundedness A mapping There is a real number Statement is bounded if s.t. Theorem 4.14: Continuous Functions Preserve Compactness Proof be metric spaces, compact If is continuous, then is also compact be an open cover of is continuous, so each of the sets is open by Theorem 4.8 is an open cover of, and is compact So there is a finite set of indices s.t. Since Statement Proof This is a finite subcover of Theorem 4.15: Applying Theorem 4.14 to be a compact metric space If is continuous, then is closed and bounded Thus, Statement is bounded See Theorem 4.14 and Theorem 2.41 Theorem 4.16: Extreme Value Theorem be a continuous real function on a compact metric space Then s.t. and Equivalently, Proof s.t.
Page 94 By Theorem 4.15, is closed and bounded So contains by Theorem 2.28 Theorem 4.17: Inverse of Continuous Bijection is Continuous Statement Proof be metric spaces, compact Suppose Define Then is continuous and bijictive by is also continuous and bijective By Theorem 4.8, it suffices to show is open in for all open sets Fix an open set in is open in compact metric space So is closed and compact by Theorem 2.35 Therefore, is a compact subset of by Theorem 4.14 So is closed in by Theorem 2.34 is 1-1 and onto, so Therefore is open
Page 95 Uniform Continuity and Compactness Friday, April 20, 2018 12:10 PM Definition 4.18: Uniform Continuity be metric spaces, is uniformly continuous on if s.t. If and, then Theorem 4.19: Uniform Continuity and Compactness Statement Proof be metric spaces, compact If is continuous, then is also uniformly continuous be given Since is continuous, s.t., so is an open cover of Since is compact, has a finite subcover So there exists finite set of points s.t. s.t. Since, Hence, s.t. By the triangle inequality and definition of, Therefore is uniformly continuous
Page 96 Theorem 4.20: Continuous Mapping from Noncompact Set Definition (a) (b) (c) Proof : If be noncompact set in Then there exists a continuous function on s.t. is not bounded is bounded but has no maximum is bounded, but is bounded is not uniformly continuous Since is noncompact, must be not closed So there exists a limit point s.t. is continuous by Theorem 4.9 is clearly unbounded is not uniformly continuous and be arbitrary Choose s.t. Taking close to We can make Since is arbitrary, but is continuous by Theorem 4.9 is bounded, since has no maximum, since, but Proof: If is not bounded establishes Example 4.21: Inverse Mapping and Noncompact Set Then But given by is continuous, and bijective is not continuous at
Page 97 Connected Set, Intermediate Value Theorem Monday, April 23, 2018 12:10 PM Definition 2.45: Connected Set and Statement Proof be a metric space, and are separated if and i.e. No point of lies in the closure of and vice versa is connected if is not a union of two nonempty separated sets Theorem 2.47: Connected Subset of Proof is connected if and only if If and, then has the following property By way of contrapositive, suppose, and s.t. and Then and are separated and Therefore is not connected By way of contrapositive, suppose is not connected Then there are nonempty separated sets and s.t.. Without loss of generality, assume. Then by Theorem 2.28, By definition of If If Statement and. So, Since and are separated, So s.t. and Then, so Theorem 4.22: Continuous Mapping of Connected Set
Page 98 Proof be metric spaces be a continuous mapping If is connected then is also connected Suppose, by way of contradiction, that Then Since is not connected, where are nonempty and separated and, where, we have Since is continuous and is closed, is also closed Therefore Since and, we have Similarly, Statement Proof So, and are separated This is a contradiction, therefore be continuous on If and if statifies Then By Theorem 2.47, By Theorem 4.22, s.t. is connected By Theorem 2.47, the result follows is connected Theorem 4.23: Intermediate Value Theorem is a connected subset of
Page 99 Derivative, Chain Rule, Local Extrema Wednesday, April 25, 2018 12:19 PM Definition 5.1: Derivative be defined (and real-valued) on is called the derivative of If is defined at point, is differentiable at If is defined, then is differentiable on Theorem 5.2: Differentiability Implies Continuity Statement Proof be defined on If is differentiable at then is continuous at Theorem 5.5: Chain Rule Statement Given is continuous on, and exists at is defined on, and is differentiable at If, then Proof is differentiable at, and By the definition of derivative, then
Page 100 If, then As So, and by continuity Definition 5.7: Local Maximum and Local Minimum be a metric space, has a local maximum at if s.t. s.t. has a local minimum at if s.t. Theorem 5.8: Local Extrema and Derivative Statement Proof If Then be defined on s.t. has a local maximum (or minimum) at if it exists By Definition 5.7, choose, then Suppose (with ), then Suppose (with ), then Therefore
Page 101 Mean Value Theorem, Monotonicity, Taylor's Theorem Friday, April 27, 2018 12:07 PM Theorem 5.9: Extended Mean Value Theorem Statement Proof Given and are continuous real-valued functions on are differentiable on Then there is a point at which, Then is continuous on and differentiable on We want to show that By definition of, we have If If Statement Proof is constant is not constant on all of s.t. By Theorem 4.16, for some, and we are done s.t. is either a global maximum or a global minimum By Theorem 5.8, Theorem 5.10: Mean Value Theorem If is continuous on and differentiable on Then Suppose s.t. in Theorem 5.9 Theorem 5.11: Derivative and Monotonicity is differentiable on If, then is monotonically increasing If, then is constant If, then is monotonically decreasing Theorem 5.15: Taylor's Theorem or
Page 102 Statement Suppose is a real-valued function on Fix a positive integer is continuous on exists, where Then between and s.t. Note Proof When, this is the Meal Value Theorem Without loss of generality, suppose Define by Then we want to show that Then Taking derivative Note that for some Define difference function Now we only need to show by by our choice of, where times on both side, we get, where disappears, since its degree is for some Therefore, Also,, by definition of By the Mean Value Theorem, for some by definition of for some After steps, for some So,
Page 103 Riemann-Stieltjes Integral, Refinement Monday, April 30, 2018 Partition A partition of a closed interval is a finite set of points 12:12 PM Definition 6.1: Riemann Integral where be a bounded real function on, for each partition of Define and to be Define the upper sum and lower sum to be where Define the upper and lower Reimann integral to be, a Well-definedness of upper and lower Riemann integral Since is bounded, s.t. Therefore for every partition of Definition 6.2: Riemann-Stieltjes Integral
Page 104 be a monotonically increasing function on be a real-valued function bouned on For each partition of, define Note This is the Riemann-Stieltjes integral of with respect to over We say is integrable with respect to with on, and write When, this is just Riemann integral Definition 6.3: Refinement and Common Refinement We say that the partition is a refinement of if Given two partitions and, their common refinement is Theorem 6.4: Properties of Refinement If is a refinement of, then
Page 105 Theorem 6.5: Properties of Common Refinement Statement Proof Outline Given 2 partitions Then Theorem 6.6 Statement Proof Outline and be the common refinement on if and only if there exists a partition s.t. If If Then s.t. Consider their common refinement