MATH 447- C13-
Introduction to Real Analysis

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

**Location: ** Altgeld
Hall 243

Grader: Mary Gramcko Tursi

Office hours: Tuesday 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.

Be prepared to accept a little absract but clarifying approach to well known, and not so well known topics related to calculus.

I: Real Numbers

Natural numbers, abelian groups, Grothendiecks construction, integers, fields, rational numbers, ordered fields, completeness, Peano's axiom, uncountability of real numbers.

II: Sequences

Limits, monotone sequences, subsequences, Bolzano-Weierstrass, limsup and liminf, application to continuous functions.

III: Metric spaces

Metric spaces, Cauchy sequences, completeness, sequential compactness and total boundedness, open, closed and compact sets, application to Heine Borel and continuity of inverses. Connectes sets, intermediate value theorem.

Spaces of continuous functions

III: Spaces of continuous functionsUniform continuity, C(K) is a complete metric space, Dini's theorem, application: interchanging differentiation and limit.

IV: Differentiation

Rolle's Lemma and Mean value theorem, application differentiation of power series.

V: Integration

Definition, interching limits, fundamental theorem, application to power series.

Practise problems for final and solutions

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

Homework (20%)

Midterm1 September 26 (25%)

Midterm2 October 26 (25%)

Final: see uncombined schedule

Video lecturehttps://uofi.box.com/s/2g32pwy5m83x3h8w5vhrmbrdm3bg3cn7

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

Additional Material: See Box

Practiceproblem 2 solutions