MATH 444-
Introduction to Real Analysis

**Time:** MWF 10:00-1:50pm

**Location: ** Altgeld
Hall 242

Grader: TBA

Office hours: Thursday 5-6

Course description :

Introduction to real analysis is a gateway. The idea is to find balance between rigorous proofs and real understanding - this principle is the core of mathematics at all levels.

Be prepared to learn to write proofs.

I: Real Numbers

What are real numbers?; sets, maps, functions, cardinality; Natural numbers, equivalence relations, integers, rational numbers, fields, ordered fields; complete ordered fields.

II: Sequences

Limits, monotone sequences, subsequences, Bolzano-Weierstrass, limsup and liminf, series.

III: Continuous Functions

Definition and permanence properties, uniform continuity, removing singularities, monotonicity, inverse function and intermediate alue theorem

IV: Differentiation

Basic rules, mean value theorem and taylor

IV: Integration

Different definition and Fundamental Theorems. Applications.

V: Power Series

Interchanging limits, examples.

VI: Misselinia

Books:

Elementary Analysis: The Theory of Calculus by Kenneth Ross, 2 edition, SpringerGrading:

Homework (40)

Midterm1 September 27

Midterm2 October 24

Midterm2 November 15 (if necessary-probably take home)

Final: see uncombined schedule

Homework1:

Homework: See folder in Box, contact me if not invited

Additional Material: See Box