Elementary Real Analysis

Elementary Real Analysis Notes

-- The following are my own notes, sourced primarily from Bartle (1976) and my own class notes. In order to follow the notes, one must understand mathematical proof. 

Review of Set Theory

The Real Numbers

Upper bounds, least upper bounds, completeness of the real numbers.

The Topology of R

Topology of Euclidean spaces: norms, the Cauchy-Schwarz inequality, open and closed sets, compactness, the Bolzano-Weierstrass and Heine-Borel theorems and connectedness.

Sequences in Euclidean spaces: Cauchy sequences, monotone sequences, pointwise, uniform and L2 convergence of sequences of functions

Continuous Functions

Local properties of continuous functions. Preservation of compactness and connectedness. Uniform continuity. Contractions and operator norm continuity of linear operators

Sequences of Functions

Sequences of continuous functions: Bernstein and Weierstrass approximation theorems.

Infinite series: Convergence and absolute convergence, convergence tests.

Series of functions: Power series, Fourier series

Useful Resources

Really good notes explaining epsilon-delta proofs

Uniform Continuity vs Pointwise Continuity

Clarifying example of how to prove a set is closed (pg4)

Textbook: Foundations of Analysis - Joseph Taylor