This is a collection of lecture notes I’ve used several times in the two-semester senior/graduate-level real analysis course at the University of Louisville. They are an ongoing project and are often updated. They are here for the use of anyone interested in such material. In return, I only ask that you tell me of mistakes, make suggestions for improvements and, when sharing with others, please give me credit or blame.

I’m very interested in feedback of any type, so don’t be shy about contacting me!

- Introduction to Real Analysis PDF file.
(Updated 19-March-2023)

- Chapter 1: Basic Ideas
- Basic set theory
- notation
- Schröder-Bernstein Theorem
- countability, uncountability
- cardinal numbers

- Chapter 2: The Real Numbers
- axioms of a complete ordered field
- basic properties of \(\mathbb{R}\)
- uncountability of \(\mathbb{R}\)

- Chapter 3: Sequences
- monotone sequences
- Cauchy sequences
- contractions

- Chapter 4: Series
- positive series
- convergence tests
- absolute convergence
- conditional convergence
- alternating series
- Riemann’s rearrangement theorem

- Chapter 5: The Topology of \(\mathbb{R}\)
- open and closed sets
- limit points
- relative topologies
- compactness
- Baire Category Theorem
- measure zero sets
- Cantor middle-thirds set

- Chapter 6: Limits of Functions
- limits
- unilateral limits
- continuity
- uniform continuity

- Chapter 7: Differentiation
- differentiation of functions
- Darboux property
- Mean Value Theorem
- Taylor’s Theorem
- l’Hôspital’s rules

- Chapter 8: Integration
- Riemann-Darboux integral
- Fundamental Theorem of Calculus.
- change of variables

- Chapter 9: Sequences of Functions
- pointwise convergence
- uniform convergence and its relation to continuity, integration and differentiation
- Weierstrass Approximation Theorem
- power series.

- Chapter 10: Fourier Series
- Dirichlet and Fejér kernels
- Cesàro convergence
- pointwise convergence.
- continuous function with divergent Fourier series
- Gibbs’ phenomenon

Contact me at Lee Larson (lee.larson@louisville.edu)