Convergence, Limits, and Continuity in Real Analysis, Lecture notes of Calculus

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An Introduction

to

Real Analysis

Cesar O. Aguilar

May 4, 2022

• 1 Preliminaries
  • 1.1 Sets, Numbers, and Proofs
  • 1.2 Mathematical Induction
  • 1.3 Functions
  • 1.4 Countability
• 2 The Real Numbers
  • 2.1 Introduction
  • 2.2 Algebraic and Order Properties
  • 2.3 The Absolute Value
  • 2.4 The Completeness Axiom
  • 2.5 Applications of the Supremum
  • 2.6 Nested Interval Theorem
• 3 Sequences
  • 3.1 Limits of Sequences
  • 3.2 Limit Theorems
  • 3.3 Monotone Sequences
  • 3.4 Bolzano-Weierstrass Theorem
  • 3.5 limsup and liminf
  • 3.6 Cauchy Sequences
  • 3.7 Infinite Series
• 4 Limits of Functions
  • 4.1 Limits of Functions
  • 4.2 Limit Theorems
• 5 Continuity
  • 5.1 Continuous Functions
  • 5.2 Combinations of Continuous Functions
  • 5.3 Continuity on Closed Intervals
  • 5.4 Uniform Continuity
• 6 Differentiation
  • 6.1 The Derivative
  • 6.2 The Mean Value Theorem
  • 6.3 Taylor’s Theorem
• 7 Riemann Integration
  • 7.1 The Riemann Integral
  • 7.2 Riemann Integrable Functions
  • 7.3 The Fundamental Theorem of Calculus
  • 7.4 Riemann-Lebesgue Theorem
• 8 Sequences of Functions
  • 8.1 Pointwise Convergence
  • 8.2 Uniform Convergence
  • 8.3 Properties of Uniform Convergence
  • 8.4 Infinite Series of Functions
• 9 Metric Spaces
  • 9.1 Metric Spaces
  • 9.2 Sequences and Limits
  • 9.3 Continuity
  • 9.4 Completeness
  • 9.5 Compactness
  • 9.6 Fourier Series
• 10 Multivariable Differential Calculus
  • 10.1 Differentiation
  • 10.2 Differentiation Rules and the MVT
  • 10.3 The Space of Linear Maps
  • 10.4 Solutions to Differential Equations
  • 10.5 High-Order Derivatives
  • 10.6 Taylor’s Theorem
  • 10.7 The Inverse Function Theorem

Preface

The material in these notes constitute my personal notes that are used in the course lectures for MATH 324 and 325 (Real Analysis I, II). You will find that the lectures and these notes are very closely aligned. The notes highlight the important ideas and examples that you should master as a student. You may find these notes useful if:

• you miss a lecture and need to know what was covered,
• you want to know what material you are expected to master,
• you want to know the level of difficulty of questions that you should expect in a test, and
• you want to see more worked out examples in addition to those worked out in the lectures.

If you find any typos or errors in these notes, no matter how small, please email me a short description (with a page number) of the typo/error. Suggestions and comments on how to improve the notes are also wel- comed.

Cesar O. Aguilar SUNY Geneseo

Preliminaries

In this short chapter, we will briefly review some basic set notation, proof methods, functions, and countability. The presentation of these topics is intentionally brief for two reasons: (1) the reader is likely familar with these topics, and (2) we include only the necessary material needed to start doing real analysis.

1.1 Sets, Numbers, and Proofs

Let S be a set. If x is an element of S then we write x ∈ S, otherwise we write that x /∈ S. A set A is called a subset of S if each element of A is also an element of S, that is, if a ∈ A then also a ∈ S. To denote that A is a subset of S we write A ⊂ S. Now let A and B be subsets of S. If A ⊂ B and B ⊂ A then A and B are said to be equal and we write that A = B. The union of A and B is the set

A ∪ B = {x ∈ S | x ∈ A or x ∈ B}

and the intersection of A and B is the set

A ∩ B = {x ∈ S | x ∈ A and x ∈ B}.

1.1. SETS, NUMBERS, AND PROOFS

Example 1.1.1. Let A and B be subsets of a set S. Show that

(A ∪ B)c^ = Ac^ ∩ Bc.

Solution. We first show that (A ∪ B)c^ ⊂ Ac^ ∩ Bc. If x ∈ (A ∪ B)c^ then by definition x /∈ (A ∪ B) and therefore x /∈ A and x /∈ B. Thus, x ∈ Ac and x ∈ Bc, that is, x ∈ Ac^ ∩ Bc. Now suppose that x ∈ Ac^ ∩ Bc, that is, suppose that x ∈ Ac^ and x ∈ Bc. Thus, x /∈ A and x /∈ B and thus x /∈ (A ∪ B). By definition, x ∈ (A ∪ B)c^ and this proves that Ac^ ∩ Bc^ ⊂ (A ∪ B)c.

We use the symbol N to denote the set of natural numbers, that is, N = { 1 , 2 , 3 , 4 ,.. .}.

The set of integers is denoted by Z so that

Z = {... , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 ,.. .}.

The set of rational numbers is denoted by

Q =

p q |^ p, q^ ∈^ Z, q^6 = 0

Notice that we have the following chain of set inclusions:

N ⊂ Z ⊂ Q.

We now review the most commonly used methods of proof. To that end, recall that a logical statement is a declarative sentence that can be unambiguously decided to be either true or false. A theorem is a logical statement that has been proved to be true using a sequence of true statements and deductive reasoning. Many theorems are usu- ally written as a conditional statement of the form “if P then Q” or

1.1. SETS, NUMBERS, AND PROOFS

symbolically “P ⇒ Q”. The statement P is called the hypothesis or assumption and Q is called the conclusion. Below are the main techniques used to prove the statement “P ⇒ Q”:

• Direct Proof: To prove the statement “P ⇒ Q”, assume that the statement P is true and show by combining axioms, defini- tions, and earlier theorems that Q is true. This should be the first method you attempt.
• Mathematical Induction: Covered in Section 1.2.
• By Contraposition: Proving the statement “P ⇒ Q” by prov- ing the logically equivalent statement “not Q ⇒ not P ”. Do not confuse this with proof by contradiction.
• By Contradiction: To prove the statement “P ⇒ Q” by con- tradiction, one assumes that “P ⇒ Q” is false and then show that some contradiction results. Assuming that “P ⇒ Q” is false is to assume that P is true and Q is false. Using these to latter as- sumptions, one attempts to derive at a contradiction of the form “R and not R”, where R is some statement. One disadvantage with proof by contradiction is that the logical contradiction that one is seeking “R and not R” is not known in advance so that the goal of the proof is unclear. Proof by contradiction frequently gets confused with proof by contraposition in the following way (do not do this): To prove that “P ⇒ Q”, assume that P is true and suppose that Q is not true. After some work using only the assumption that Q is not true you show that P is not true and thus you say that there is a contradiction because you assumed that P is true. What you have really done is proved the contra- positive statement. Thus, if you believe that you are proving a

1.1. SETS, NUMBERS, AND PROOFS

“P ⇔ Q”, is called a biconditional statement. Thus, to prove that the biconditional statement “P ⇔ Q” is true one must prove that both “P ⇒ Q” and “Q ⇒ P ” are true.

Example 1.1.3. Let A, B, and C be subsets of a set S. Prove that (A ∪ B) ⊂ C if and only if A ⊂ C and B ⊂ C.

1.1. SETS, NUMBERS, AND PROOFS

Exercises

Exercise 1.1.1. Let A and B be subsets of a set S. Show that A ⊂ B if and only if Bc^ ⊂ Ac

Exercise 1.1.2. Find the power set of S = {x, y, z, w}.

Exercise 1.1.3. Let A = {α 1 , α 2 , α 3 } and let B = {β 1 , β 2 }. Find A × B.

Exercise 1.1.4. Let x ∈ Z. Prove that if x^2 is even then x is even. Do not use proof by contradiction.

Exercise 1.1.5. Prove that if x and y are even natural numbers then xy is even. Do not use proof by contradiction.

Exercise 1.1.6. Prove that if x and y are rational numbers then x + y is a rational number. Do not use proof by contradiction.

Exercise 1.1.7. Let x and y be natural numbers. Prove that x and y are odd if and only if xy is odd. Do not use proof by contradiction.

1.2 Mathematical Induction

therefore S = N.

Mathematical induction is frequently used to prove formulas or in- equalities involving the natural numbers. For example, consider the validity of the formula

1 + 2 + 3 + · · · + n = 12 n(n + 1) (1.1)

where n ∈ N. In words, the identity (1.1) says that the sum of all the integers from 1 to n equals 12 n(n + 1). We use induction to show that this formula is true for all n ∈ N. Let S be the subset of N consisting of the natural numbers that satisfy (1.1), that is,

S = {n ∈ N | 1 + 2 + 3 + · · · + n = 12 n(n + 1)}.

If n = 1 then (^12) n(n + 1) = 12 (1 + 1) = 1.

Thus, 12 n(n + 1) is equal to the sum of all the integers from 1 to n = 1. Hence, (1.1) is true when n = 1 and thus 1 ∈ S. Now suppose that some k ∈ N satisfies (1.1), that is, suppose that k ∈ S. Then we may write that 1 + 2 + · · · + k = 12 k(k + 1). (1.2)

We will prove that the integer k + 1 also satisfies (1.1). To that end, adding k + 1 to both sides of (1.2) we obtain

1 + 2 + · · · + k + (k + 1) = 12 k(k + 1) + (k + 1).

Now notice that we can factor (k + 1) from the right-hand side and through some algebraic steps we obtain that

1 + 2 + · · · + k + (k + 1) = 12 k(k + 1) + (k + 1) = (k + 1)[^12 k + 1] = 12 (k + 1)(k + 2).

1.2. MATHEMATICAL INDUCTION

Hence, (1.1) also holds for n = k + 1 and thus k + 1 ∈ S. We have therefore proved that S satisfies properties (i) and (ii), and therefore by mathematical induction S = N, or equivalently that (1.1) holds for all n ∈ N.

Example 1.2.3. Use mathematical induction to show that 2n^ ≤ (n+1)! holds for all n ∈ N.

Example 1.2.4. Let r 6 = 1 be a constant. Use mathematical induction to show that

1 + r + r^2 + · · · + rn^ =^1 −^ r

n+ 1 − r holds for all n ∈ N.

Example 1.2.5 (Bernoulli’s inequality). Prove that if x > −1 then (1 + x)n^ ≥ 1 + nx for all n ∈ N.

Proof. The statement is trivial for n = 1. Assume that for some k ∈ N it holds that (1 + x)k^ ≥ 1 + kx. Since x > −1 then x + 1 > 0 and therefore

(1 + x)k(1 + x) ≥ (1 + kx)(1 + x)

= 1 + (k + 1)x + kx^2

≥ 1 + (k + 1)x.

Therefore, (1 + x)k+1^ ≥ 1 + (k + 1)x, and the proof is complete by induction.

There is another version of mathematical induction called the Prin- ciple of Strong Induction which we now state.

1.2. MATHEMATICAL INDUCTION

Exercises

Exercise 1.2.1. Prove that n < 2 n^ for all n ∈ N.

Exercise 1.2.2. Prove that 2n^ < n! for all n ≥ 4, n ∈ N.

Exercise 1.2.3. Use induction to prove that if S has n elements then P(S) has 2n^ elements. Hint: If S is a set with n + 1 elements, for instance S = {x 1 , x 2 ,... , xn, xn+1}, then argue that P(S) = P( S˜) ∪ T where S˜ = S{xn+1} and T consists of subsets of S that contain xn+1. How many sets are in P( S˜) and how many are in T? And what is P( S˜) ∩ T? Explain carefully.

1.3 Functions

1.3 Functions

Let A and B be sets. A function from A to B is a rule that assigns to each element x ∈ A one element y ∈ B. The set A is called the domain of f and B is called the co-domain of f. We usually denote a function with the notation f : A → B, and the assignment of x to y is written as y = f (x). We also say that f is a mapping from A to B, or that f maps A into B. The element y assigned to x is called the image of x under f. The range of f , denoted by f (A), is the set

f (A) = {y ∈ B | ∃ x ∈ A, y = f (x)}.

In the above definition of f (A), we use the symbol ∃ as a short-hand for “there exsits”. By definition, f (A) ⊂ B but in general we do not have that f (A) = B, in other words, the range of a function is generally a strict subset of the function’s co-domain.

Example 1.3.1. Consider the mapping f : Q → Z defined by

f (x) =

1 , x ≥ 0 − 1 , x < 0.

The image of x = 1/2 under f is f (1/2) = 1. The range of f is f (Q) = { 1 , − 1 }.

Example 1.3.2. Consider the function f : N → P(N) defined by

f (n) = { 1 , 2 ,... , n}.

The set S = { 2 , 4 , 6 , 8 ,... , } of even numbers is an element of the co- domain P(N) but is not in the range of f. As another example, the set N ∈ P(N) itself is not in the range of f.

Function’s whose range is equal to it’s co-domain are given a special name.