Course Description
MTH 435 and its second part MTH 436 provide an introduction to the wonderful world of real numbers,
real analysis and topology. Real analysis is the precise mathematical theory behind Calculus.
The emphasis will be on mathematical precision and developing strong skills in proving theorems.
You will see complete proofs done in class and learn how to write your own proofs. At the same time,
you will gain a deep understading of the fundamental concepts of Calculus
that you already know from the intuitive point of view. Those include limits, continuity, sequences of numbers and differentiation.

Many theorems in analysis that we will prove are particular cases of more general theorems
in metric spaces. Therefore, we will study metric spaces which belong to the area of mathematics called topology.

Current Downloads

You will find here current downloads: solutions to homework problems, additional class notes etc.

Homework 1 Solutions
Homework 2 Solutions
Homework 3 Solutions
Homework 4 Solutions

Homework 5 Solutions
Homework 6 Solutions
Homework 7 Solutions
Homework 8 Solutions
Homework 9 Solutions

Exams and Evaluation

**There will be two exams during the semester, Friday Oct 17, 3-5 and Friday Nov 21, 3-5, Lippitt 204.**
Our **Final is scheduled (by the Office of Enrollment Services) on Fri, Dec 12, 3-6pm, Lippitt 204**.
Final Exam Schedule.

Your grade will be based upon a possible total of 550 points, as
follows:

- Exam 1 -- 100 points,
- Exam 2 -- 100 points,
- Homework -- 150 points,
- Final exam -- 200 points,

Weekly homework will be assigned in class.

Topics

INTRODUCTION

Mathematical Logic

Mathematical Induction

REAL NUMBERS

Axioms, Ordering

The Completeness Axiom

Density of Rationals and Irrationals

Countable and Uncountable Sets

SEQUENCES OF REAL NUMBERS

Intervals, Neighborhoods, the Triangle Inequality

Convergence, Divergence, Limits

Cauchy Sequences

Subsequences

The Bolzano-Weierstrass Theorem

REAL FUNCTIONS

Limits

Continuity

Boundedness Theorem

Min-Max Theorem

Intermediate Value Theorem

Uniform Continuity

The Derivative

Differentiability

Proofs of the Product, Quotient, Chain Rules

Local Extrema

Mean Value Theorem

Monotonicity and the Derivative

METRIC SPACES

Definition

Open and Closed Sets

The Real Line as a Metric Space

Sequences in a Metric Space

Completeness

Compactness

Continuous Functions