# Real Analysis

## Calendar

LEC # TOPICS KEY DATES
1 Sets, Set Operations, and Mathematical Induction
2 Cantor’s Theory of Cardinality (Size) Assignment 1 due
3 Cantor’s Remarkable Theorem and the Rationals’ Lack of the Least Upper Bound Property
4 The Characterization of the Real Numbers Assignment 2 due
5 The Archimedian Property, Density of the Rationals, and Absolute Value
6 The Uncountabality of the Real Numbers Assignment 3 due
7 Convergent Sequences of Real Numbers
8 The Squeeze Theorem and Operations Involving Convergent Sequences Assignment 4 due
9 Limsup, Liminf, and the Bolzano-Weierstrass Theorem
10 The Completeness of the Real Numbers and Basic Properties of Infinite Series Assignment 5 due
11 Absolute Convergence and the Comparison Test for Series
12 The Ratio, Root, and Alternating Series Tests Assignment 6 due
Midterm Exam Midterm Exam due
13 Limits of Functions
14 Limits of Functions in Terms of Sequences and Continuity Assignment 7 due
15 The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet’s Function
16 The Min/Max Theorem and Bolzano’s Intermediate Value Theorem Assignment 8 due
17 Uniform Continuity and the Definition of the Derivative
18 Weierstrass’s Example of a Continuous and Nowhere Differentiable Function Assignment 9 due
19 Differentiation Rules, Rolle’s Theorem, and the Mean Value Theorem
20 Taylor’s Theorem and the Definition of Riemann Sums Assignment 10 due
21 The Riemann Integral of a Continuous Function
22 The Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula Assignment 11 due
23 Pointwise and Uniform Convergence of Sequences of Functions
24 Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits Assignment 12 due
25 Power Series and the Weierstrass Approximation Theorem
Final Assignment Final Assignment due

Fall 2020
Lecture Notes
Lecture Videos
Exams
Problem Sets