Course Topics

Topics in bold are the Headline theorems we will prove.

Compeleteness

1. Properties of the Reals
2. Completeness Axiom
3. Suprema and Infima

Sequences

1. Notion of a limit.
2. Proof of limit laws
3. Cauchy sequences
4. Bolzano-Weierstrass Theorem

Series

1. Definition of the limit of a series.
2. Use convergence tests to show whether a series converges.
3. Convergence Tests.

Metric Spaces

1. Definition.
2. Sequences in a metric space and their convergence
3. Equivalent metrics

Functions

1. Limits and Continuity
2. Extreme Value Theorem
3. Intermediate Value Theorem

By the end of the course, you should be able to cope with abstract mathematical ideas and observe how they allow the proof of general results. For many of you, this course is the culmination of your Mathematics studies, where you combine ideas from as far back as Calculus to techniques from MTH307 to rigorously prove some fundamental mathematical results.