Real Analysis 2004

Information about this course can be found here on the Pure Mathematics web page.

Email me by clicking here.

My consulting time for this course is 1.30-2.30 on Thursdays. If you cannot come at that time email or ring me to make an appointment.

Tutorials at 2.00 Monday in Room 1.08 and 3.00 Monday in Room G05 in Weeks 3, 5, 7, 9, 11 and 13.

There will be only one lecture in Week 13 on Monday the 7th June at 3.00 in G05 when I will revise and talk about the examination.

Handouts

Class Exercises and Tutorial Exercises

Course guide

All references are to the textbook. Real Analysis. A First Course Russell A.Gordon

Week 1:
- 1.1 What is a real number ?
- 1.2 Absolute Value, Interval and Inequalities. Omitted Lemma 1.11, Theorem 1.12, Theorem 1.13.
- 1.3 The Completeness Axiom
Week 2:
- Finish 1.3.
- 1.4 Countable and Uncountable Sets.
- 1.5 Real functions. This adds nothing to what you know already from Maths 1 so I will omit it but assume you know it.
- 2.1 Convergent Sequences up to Theorem 2.5
Week 3:
- Tutorials on Monday. Room 108 at 2.00 and Room G08 at 3.00.
- 2.1 Finish convergent sequences starting with Theorem 2.5. Omit Def 2.6
- 2.2 Monotone Sequences and Cauchy Sequences
Week 4:
- Finish Monotone Sequences and Cauchy Sequences
- 2.3 Subsequences. Omit Def 2.20, Thm 2.21 and Thm 2.22.
- 3.1 The Limit of a Function
Week 5:
- Finish 3.1 The Limit of a Function
- 3.2 Continuous Functions
- 3.3 Intermediate and Extreme Values
Week 6:
- 3.4 Uniform Continuity. Omit Thm 3.29, Thm 3.30.
- Omit 3.5 Monotone Functions
- 4.1 The Derivative of a Function.
Week 7:
- 4.2 The Mean Value Theorem. Omit Thm 4.15, 4.16 4.17 and 4.18
- Omit 4.3 Further Topics on Differentiation.
Week 8:
- 5.1 The Riemann Integral
- 5.2 Conditions for Riemann Integrability
Week 9:
- More about 5.2 Conditions for Riemann Integrability
- 5.3 The Fundamental Theorem of Calculus
Week 10:
- More about 5.2 Conditions for Riemann Integrability
- 5.3 The Fundamental Theorem of Calculus
Week 11:
- More about 5.3 The Fundamental Theorem of Calculus
- Omit 5.4 and 5.5.
- 6.1 Convergence of infinite series
- 6.2 The Comparison Test. Omit Thm 6.7.
Week 12: