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
|
|
Bolzano-Weierstrass theorem
|
5
|
|
Open sets, closed sets, bounded sets, and
compact sets in R
|
6
|
|
Limits of real-valued 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
|
