Course Description:
Real numbers (order, absolute value, bounded subsets, completeness
property, Archimedean property). Supremum and infimum. Sequences (limit, Cauchy
sequence, recurrence sequence, increasing, decreasing sequence). Topology of R.
Functions (limit, continuity at a point, continuity on an interval). Uniform
continuity (on an interval) and relation between continuity and uniform
continuity. Differentiability and relation between differentiability and
continuity. Rolle’s theorem. Mean value theorem with applications. Taylor theorem
with remainder. Riemann–Stieltjes integral. Fundamental theorem of calculus.
Textbook:
Introduction to real analysis, by Robert Bartle and Donald
Sherbert.
References:
1) Principles of Mathematical
Analysis, by Walter Rudin.
2) Mathematical Analysis,
Second Edition, by Tom Apostol.
Course Outline and Schedule:
The following is a rough plan. As the
course progresses, I may include new topics and/or delete some of the
ones listed here.
Topic

Week

The completeness property of R

1

The Archimedean principle in R

1

Limit of a sequence

2

Convergent sequences

2

Monotone and bounded sequences

3

Cauchy sequences

3

Subsequences

4

BolzanoWeierstrass theorem

5

Open sets, closed sets, bounded sets, and
compact sets in R

6

Limits of realvalued functions

7

Definition of limits by neighborhoods

7

Definition of limits by sequences

8

Continuous functions on R

9

Sequence definition and neighborhood
definition of continuity

9

Boundedness of continuous functions on compact
intervals

10

The extreme value theorem

11

The intermediate value theorem

11

Uniformly continuous functions

12

The sequential criterion for uniform
continuity

12

The derivative of functions

13

Role’s theorem

13

Mean value theorem

13

Generalized mean value theorem

14

Taylor theorem with remainder

14

L’Hospital’s rule

14

Riemann–Stieltjes integral

15

Fundamental theorem of calculus

16
