Real analysis

Builds on

Required prerequisites, all the way down: 35

You will learn to

• Prove the Bolzano–Weierstrass theorem and the intermediate and extreme value theorems from completeness.
• Distinguish pointwise from uniform convergence and prove that uniform limits of continuous functions are continuous.
• Prove the fundamental theorem of calculus for the Riemann integral.

Definition

• aₙ → L if for every ε > 0 there is N such that |aₙ − L| < ε for all n ≥ N.
• fₙ → f uniformly on S if sup_{x∈S} |fₙ(x) − f(x)| → 0.
• A sequence is Cauchy if for every ε > 0 there is N such that |aₘ − aₙ| < ε for all m, n ≥ N.

Theorem

• In ℝ, a sequence converges if and only if it is Cauchy.
• Bolzano–Weierstrass: every bounded sequence of real numbers has a convergent subsequence.
• Uniform limits of continuous functions are continuous.
• There are continuous functions that are differentiable nowhere (Weierstrass).

Common misconceptions

• Interchanging limits without justification; a pointwise limit of continuous functions need not be continuous.
• Thinking every bounded function on [0, 1] is Riemann integrable; the indicator function of ℚ is not.

Leads to

Optional discoveries

• Formalising mathematics in Lean and Mathlib Mathlib formalises much of undergraduate analysis, so theorems such as these can be stated and checked in Lean.
• ← Sequences and series
• ← Introductory calculus
• ← Constructing ℤ, ℚ and ℝ
• ← Completeness, compactness and Löwenheim–Skolem

Learning pathways

• Undergraduate mathematics major
• Graduate preparation

Further reading

• Walter Rudin, Principles of Mathematical Analysis (1953)
• Stephen Abbott, Understanding Analysis (2001)