# Analysis I

CU Boulder

Instructor: Yuhao Hu

Email: yuhao.hu@colorado.edu

Office: Math 225

Office Hours: WF 4:00-5:45pm

Lectures: MWF 9:00-9:50am at ECCR 110

## Overview

In some sense, mathematical analysis was introduced as a new language of mathematics in order
to bring rigor to the subject of calculus (e.g., to understand the meaning of "limit").
The development of this 'language', like the invention of a telescope,
allows one to make more accurate observations,
put forward precise questions and propositions, and construct useful theories and techniques.
These new questions, theorems, etc., in turn,
enrich not only the language but also the body of mathematics that's tied to it.
Almost 200 years after Cauchy's 1821 book *Cours d'analyse*,
analysis is now a 'universe' in itself.

This course is a beginning step towards understanding analysis.
We start with a bit of set theory, followed by
an introduction to the structure of
real numbers. We develop language/tools that allow
us to make sense of limit, which further allows us
to take a more precise view of functions and their
continuity. This course ends with basic theories of differentiation
and integration.

## Textbook

**T1.**

*Understanding Analysis, 2nd ed.*by Stephen Abbott

**T2. (Optional)**

*Analysis I, 3rd ed.*by Terence Tao

## Schedule

(weekly, only the first meeting of the week is dated; "S" stands for "Section" in**T1**)

**01/13**Number systems, sets (S1.1); functions, logic (S1.2)

**01/22**Characterization of $\mathbb{R}$, axiom of completeness (S1.3); Consequences of completeness (S1.4)

**01/27**Cardinality (S1.5); Cantor's theorem (S1.6); puzzles arising from infinite sums (S2.1)

**02/03**Limit of a sequence (S2.2);

**02/10**

**02/17 (Midterm I on Friday)**

**02/24**

**03/02**

**03/09**

**03/16**

**03/23-27 (Spring break; no class)**

**03/30**

**04/06 (Midterm II on Friday)**

**04/13**

**04/20**

**04/27**

## Homework

(Due on each**Friday**, unless specified otherwise.)

**Homework 1**(Lectures 1-2) due Jan. 24

[Lecture 1]

**Reading:**pp. 1-7 (note: The default textbook is

*'Understanding Analysis'*.)

**Exercises:**[p. 11, Sec. 1.2: 1, 2, 3(a,c)]

[Lecture 2]

**Reading:**pp. 7-11

**Exercises:**[p. 12, Sec. 1.2: 6(a,b), 8, 10, 11, 12]

## Exams

**Midterm I:**2/21 Friday

**Midterm II:**4/10 Friday

**Final:**TBD

## Grading

**Homework:**30%

**Midterm I:**20%

**Midterm II:**20%

**Final Exam:**30%