Introduction to Real Analysis

Why Rigorous Analysis?

About This Chapter

In this introductory chapter, we explore why it is necessary to rigorously revisit calculus. The concepts of limits and continuity learned in high school actually contain ambiguities. By seeing examples where these ambiguities cause problems, the goal is to appreciate the need for precise definitions.

Prerequisites

  • High-school-level differentiation and integration
  • Limits of sequences (intuitive understanding)
  • Reading graphs of functions

Table of Contents

1. Why Rigor Is Needed

Examples where intuition alone fails.

  • The ambiguity of "approaching infinitely close"
  • The trap of infinitesimals
  • Historical confusion and resolution

2. What Are the Real Numbers?

"Gaps" on the number line.

  • Why rationals are not enough
  • Does $\sqrt{2}$ truly exist?
  • Completeness of the reals (intuition)

3. Limits of Sequences

What "approaching" really means.

  • Intuitive picture of convergence
  • Arbitrarily close
  • Divergence and oscillation

4. Supremum and Infimum

When the "maximum" does not exist.

  • Sets bounded above
  • The concept of supremum
  • Importance of the least-upper-bound axiom

5. Extended Reals and Arithmetic with Infinity

Arithmetic involving $\pm\infty$.

  • Definition of the extended real line
  • Arithmetic rules at a glance
  • Seven indeterminate forms and examples

6. Intuition of Continuity

What "the graph is connected" means.

  • Examples of discontinuity
  • Intuitive properties of continuous functions
  • Intermediate Value Theorem (intuitive version)

7. Motivation for Rigor

Why we refine definitions.

  • Weierstrass's monster
  • Examples where intuition fails
  • Preparing for the $\varepsilon$-$\delta$ definition

Illustration: Gaps in the Rationals

0 1 2 3
$\sqrt{2} \approx 1.414$
Not in the rationals!
The rationals $\mathbb{Q}$ have "gaps" — fill them with the reals $\mathbb{R}$
Figure 1: The rational number line has a "gap" at the position of $\sqrt{2}$

Illustration: Convergence of a Sequence

n aₙ L L+ε L−ε N For n ≥ N, |aₙ − L| < ε
Figure 2: Schematic of sequence convergence (all terms lie within the ε-band for $n \geq N$)

Key Concepts

Completeness of the Reals (Intuitive Version)

The number line has no "gaps." The rationals alone leave a hole at the position of $\sqrt{2}$, but the reals fill every such hole.

Existence of the Supremum (Intuitive Version)

Every nonempty set of real numbers that is bounded above has a least upper bound (supremum). This property does not hold for the rationals and is an essential characteristic of the reals.

Convergence of a Sequence (Intuitive Version)

A sequence $\{a_n\}$ converges to $L$ if $a_n$ can be made arbitrarily close to $L$ by taking $n$ large enough. In other words, from some index onward, all terms are near $L$.

Illustration: Supremum

sup A upper bounds a b Set A = [a, b) sup A = least upper bound = b (not in A)
Figure 3: The supremum of the set $A = [a, b)$

When Intuition Fails

$0.999\ldots = 1$ ?

$0.999\ldots$ is equal to $1$. This is not "infinitely close to 1" but truly equal. Without a rigorous definition of limits, this fact is a common source of confusion.

Continuous but Nowhere Differentiable

Weierstrass constructed a function that is continuous everywhere but differentiable nowhere. Being "smoothly connected" and being "differentiable" are not the same thing.

Interchanging Limits

$\displaystyle\lim_{n \to \infty} \lim_{m \to \infty} a_{n,m}$ and $\displaystyle\lim_{m \to \infty} \lim_{n \to \infty} a_{n,m}$ are not equal in general. Knowing the conditions under which they can be exchanged requires rigorous theory.

Gaps in the Rationals

Considering only the rationals, the set $\{x \in \mathbb{Q} : x^2 < 2\}$ has no largest element. Its supremum does not exist within the rationals. This is why we need the real numbers.

What You Can Understand at This Level

The Meaning of Limit Calculations

$\displaystyle\lim_{n \to \infty} \frac{1}{n} = 0$ means not just that "$1/n$ gets smaller and smaller," but that it can be made smaller than any given positive number.

Intuition of the Intermediate Value Theorem

If a continuous function $f$ satisfies $f(a) < 0 < f(b)$, then there exists a $c$ with $f(c) = 0$. This is a consequence of "the graph being connected."

Handling Infinity

Preparing to handle "approaching infinity" and "infinitely large" rigorously. While $\infty$ is not a number, understanding what it means as notation in limits.

Study Tips

  • Keep asking why: Continuously question "why is this correct?"
  • Look for counterexamples: Try to find cases where intuitive claims fail
  • Question the language: Notice the ambiguity in phrases like "approaching infinitely close" or "infinitely small"
  • Learn the history: Understanding the struggles of 18th–19th century mathematicians clarifies the motivation