## Mathematics 225

### Analysis I

Credits: 3

Length of Course: 14 weeks

Classroom Hours per Week: 4

Prerequisites: Mathematics 114 and 12 credits, and Mathematics 120 or Mathematics 215.

Text: Analysis with an Introduction to Proof, Steven R. Lay, latest edition

### Course Description:

A first course in Pure Mathematics, introducing the ideas of limits, convergence and divergence of sequences. This course is designed for students majoring in Mathematics, Science, Engineering, and Theoretical Computer Science.

### Course Outline:

Length Description

Week 1

Logic and Proof: Logical Connectives, Quantifiers, Techniques of Proof I

Week 2

Logic and Proof: Techniques of Proof II

Sets and Functions: Basic Set Operations, Relations

Week 3

Sets and Functions: Functions, Cardinality, Axioms of Set Theory

Week 4

The Real Numbers: Natural Numbers and Induction, Ordered Fields, The Completeness Axiom

Week 5

The Real Numbers: Topology of the Real Numbers, Compact Sets

Sequences: Convergence

Week 6

Sequences: Limit Theorems, Monotone Sequences and Cauchy Sequences, Subsequences

Week 7

Limits and Continuity: Limits of Functions, Continuous Functions, Properties of Continuous Functions

Week 8

Limits and Continuity: Uniform continuity

Differentiation: The Derivative, The Mean Value Theorem

Week 9

Differentiation: L’Hopital’s Rule, Taylor’s Theorem

Integration: The Riemann Integral

Week 10

Integration: Properties of the Riemann Integral, The Fundamental Theorem of Calculus

Infinite Series: Convergence of Infinite Series

Week 11

Infinite Series: Convergence Tests, Power Series

Sequences and Series of Functions: Pointwise and Uniform Convergence

Week 12

Sequences and Series of Functions: Applications of Uniform Convergence, Uniform Convergence of Power Series

Week 13

Review

Week 14

Final Exam

### Evaluation:

10-20% Quizzes and Homework Midterm Exam(s) Final Exam

### Instructors

Arman Ahmadieh B.Sc., M.Sc. (Sharif University of Technology)
Hayri Ardal, B.Sc.(Bogazici), Ph.D.(Simon Fraser)
Kim Peu Chew, B.Sc. (Nanjing), M.A., Ph.D. (British Columbia)
Ana Culibrk, B.Sc.,M.Sc. (Belgrade),M.Sc.(British Columbia)
Rika Dong, B.Sc. (Simon Fraser), M.Sc. (Regina)
Sam Ekambaram, B.Sc., M.Sc. (Madras), M.Sc., Ph.D. (Simon Fraser)
Himadri Ganguli, B.Sc., M.Sc. (Chennai), Ph.D. (Simon Fraser)
Peter Hurthig, B.Sc., M.Sc. (British Columbia)