About
Welcome to your first Real Analysis course. In this course, we will go back to Calculus 1 and 2 and redo almost everything, but this time with proofs . By the end of the course you will (ideally) not be afraid of ε-δ proofs, uniform continuity, or Riemann sums, but you may be afraid of Cantor sets. We shall see. A slightly more official version of the contents of the course can be found here.
General information
When? MonWedFri 12:00pm-12:50pm.Where? Kirwan 0103.
Who? Aris Papadopoulos (just call me Aris: "R"+"iss" or "air"+"iss").
How? That's a more multifaceted question. Here are some possibilities:
- How to reach me?
{aris}@{umd}.{edu}(without{}, obviously). - How to study? Come to lectures and work through the HWs.
- How to get more help? Come to my office hour (Fridays 11:00am at Kirwan 4311).
Resources
We will be followingBaby Rudinrather closely. The book can be found here. It is not the easiest of reads, so I will also be providing additional notes if we need them. Other good sources include Stephen Abbott's
Understanding Analysis, and (I am told but haven't checked myself) Patrick M. Fitzpatrick's
Advanced Calculus.
Practice Midterm - Questions
Midterm - Questions and answers
Course Log
Week 1 - Irrational Numbers. The least upper bound property, ordered fields. The Archimedean property. (Rudin pp.1-9)Week 2 - Existence of n-th roots. (Rudin pp.9-10); Countable and Uncountable sets. Metric Spaces. (Rudin pp.24-32)
Week 3 - Metric Spaces. Compact Sets. (Rudin pp.30-37)
Week 4 - Relative openness (Extra notes). Compact Sets. (Rudin pp. 37-41) Connected Sets (in HW3).
Week 5 - Perfect sets (in HW4). Heine-Borel. Sequences.
Week 6 - Convergence (Extra notes). Subsequences. Cauchy Sequences. (Rudin pp. 47-54)
Week 7 (Midterm week) - Monotone Convergence Theorem. Upper and Lower Limits. Some Special Sequences. (Rudin pp. 55-58)
Week 8 - Series. (Rudin pp. 57-70)
Week 9 - Continuity.
Week 10 - Uniform Continuity. The Extreme and Intermediate Value theorems.
Week 11 - Differentiation (Extra notes).
Week 12 - L'Hôpital's Rule. Integration.
Homeworks
HW 1 (here, due 02/09, beginning of class) - Typo fixed 02/06.HW 2 (here, due 02/16, beginning of class) - Updated 02/11.
HW 3 (here, due 02/23, beginning of class) - Typos fixed 02/17 and 02/18.
HW 4 (here, due 03/02, beginning of class) - Typo fixed 02/25.
HW 5 (here, due 03/09, beginning of class)
HW 6 (here, due 04/01, beginning of class)
HW 7 (here, due 04/08, beginning of class)
HW 8 (here, due 04/15, beginning of class) - Typo fixed 04/10.
HW 9 (here, due 04/22, beginning of class) - Typo fixed 04/20.
HW 10 (here, due 04/29, beginning of class)