# Winter Quarter 2006, Math CS120, Introduction to Analysis

Instructor:
A. Agboola
Lecture:
TuTh 9:30am-11:00am, Bldg. 494, Room 148
There will also be a regular problem session. Details concerning this will be announced later.
Office:
6724 South Hall, (805) 893-3844
Office hours:
TuTh 11:15am-12:30pm
Textbooks:
J. C. Burkhill, A first course in mathematical analysis, CUP (1978) (required).

Homework:
The following link will take you to the homework assignments:

# Homework Assignments

Homework will be assigned at regular intervals, and some of it will be collected. Homework assignments should be written out {\bf neatly} on letter-size paper. Please write on only one side of each sheet of paper, and please begin each answer to a new quaetion on a fresh page.

Your final grade on the course will be determined by the quality of the homework that you hand in. No examinations will be given in this course.
Course Outline:
We shall aim to cover the following topics. Additional topics will be covered if time permits.

(1) Introduction: The need for a rigorous treatment of analysis.

(2) Numbers: Review of basic algebraic and order properties of the real numbers. Algebraic properties of inequalities. Upper and lower bounds. Statement, as an axiom, that a non-empty bounded set of real numbers has a least upper bound. Complex numbers. Schwartz and triangle inequalities.

(3) Limits and Series: Limits; definitions and basic properties. $O$ and $o$ notation. Monotonic sequences. Upper and Lower limits. General principle of convergence for real and complex sequences.
Convergence of series; basic ideas; standard examples, from first principles. Simple comparison tests. Power series; radius of convergence. Multiplication of absolutely convergent series. Conditionally convergent series; partial summation.

(4) Functions of a real variable; limits and continuity: Definition of a limit (in terms of $\varepsilon$ and $\delta$) and of continuity at a point. Elementary properties (sum, product, quotient; composition of continuous functions).
Functions continuous on a closed interval. Intermediate value theorem; inverse functions. Bolzano-Weierstrass theorem. Uniform continuity, existence and attainment of bounds.