# Real Analysis I with Prof. Su

### Course Description

This course is a rigorous analysis of the real numbers, as well as an introduction to writing and communicating mathematics well. Topics will include: construction of the real numbers, fields, complex numbers, topology of the reals, metric spaces, careful treatment of sequences and series, functions of real numbers, continuity, compactness, connectedness, differentiation, and the mean value theorem, with an introduction to sequences of functions.

The mean value theorem: given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints (Source: Wikipedia).
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### Video Lectures & Study Materials

Visit the official course website for more study materials: https://www.math.hmc.edu/~su/math131/

# Lecture Play Lecture
I. Introduction to Set Theory
1 Constructing the Rational Numbers (1:02:06) Play Video
2 Properties of ℚ (1:07:38) Play Video
3 Construction of the Reals (1:15:29) Play Video
4 The Least Upper Bound Property (1:12:05) Play Video
5 Complex Numbers (1:07:57) Play Video
6 Principle of Induction (1:11:58) Play Video
II. Countability & Cardinality
7 Countable and Uncountable Sets (1:13:16) Play Video
8 Cantor Diagonalization and Metric Spaces (1:16:28) Play Video
9 Limit Points (1:12:11) Play Video
III. Sets and Compactness
10 The Relationship Between Open and Closed Sets (1:13:02) Play Video
11 Compact Sets (1:10:54) Play Video
12 Relationship of Compact Sets to Closed Sets (1:15:55) Play Video
13 Compactness and the Heine-Borel Theorem (1:09:29) Play Video
IV. Connectedness and Spaces
14 Connected Sets, Cantor Sets (1:03:52) Play Video
15 Convergence of Sequences (59:00) Play Video
16 Subsequences, Cauchy Sequences (1:08:01) Play Video
17 Complete Spaces (1:05:23) Play Video
18 Series (1:11:06) Play Video
19 Series, Convergence Tests, Absolute Convergence (1:02:58) Play Video
V. Continuity & Functions
20 Functions - Limits and Continuity (1:04:26) Play Video
21 Continuous Functions (48:19) Play Video
22 Uniform Continuity (1:10:23) Play Video
23 Discontinuous Functions (44:20) Play Video
VI. Differentiation
24 The Derivative and the Mean Value Theorem (1:05:22) Play Video
25 Taylor's Theorem, Sequence of Functions (1:09:43) Play Video
26 Ordinal Numbers and Transfinite Induction (1:00:51) Play Video