This syllabus will be filled out more as the semester progresses. Check back often for updates and extras.
I expect the reading to be done prior to coming to class, and the exercises to be started.
Lesson 1
On day 1, we will look at the "Big Picture" of mathematics, and how real analysis fits in. We'll also look at some of the surprising results in store for us this semester, and one of my favorite proofs.
Think About: What are the major themes of mathematics? major techniques?
Lesson  Date  Section  Reading  Assignments  Highlights & Extras 

1  14Jan  1.1: Preliminaries  pp. 110  Ch1: 1,3,5  my favorite proof 
Block I: Real Numbers
In the first block, we will review proof techniques and learn about how to construct the real numbers starting with the natural numbers 0,1,2,… We will also take a look at how the reals are used to construct Euclidean space.
Think About: How many numbers are there? Are there more real numbers than rational numbers? Why do you think so? What does continuity really mean? What are some realworld situations in which the notion of continuity (or discontinuity) appears?
Lesson  Date  Section  Reading  Assignments  Highlights & Extras 

2  16Jan  1.1: Preliminaries  pp. 110  Ch1: 6,7  Hilbert and mathematical rigor (Skill: proof writing) 
3  18Jan  1.2: Cuts  pp. 1017  Ch1: 9,10,11,12,14,15,17  construction of the Reals 
4  23Jan  1.2: Cuts  pp. 1721  [Turnin: 1,3,5,6,7,9!,10!]  Dedekind 
5  25Jan  1.2: Cuts  (Fun Math: geometric spaces)  
6  29Jan  1.3: Euclidean Space  pp. 2127  Ch1: 18,26 [Turnin: 11,14!,15!,18] 

7  31Jan  1.6: Continuity  pp. 3639  Ch1: 19,38  
8  4Feb  1.6: Continuity  [Turnin: these problems]  points of discontinuity  
9  6Feb  1.4: Cardinality  pp. 2832  Ch1: 25,32  Cantor & the uncountability of the Reals 
10  8Feb  Applications/Review  [Discuss: 19,38]  (Application: cake cutting) The Fair Division Calculator 

11  12Feb  WPR I  about WPR I  [Turnin: 19,38] 
Block II: Sets & Sequences (Topology)
In this block, we examine series and sequences of real numbers, and answer the question of when a limit of these numbers exists. We also look at what happens when sets are equipped with a notion of distance. Much of the required terminology comes from the area of Topology, which is essentially a mathematical study of the "closeness" of points.
Think About: What makes a sequence of points converge? Is it possible for a sequence to converge to more than one number? What if you're allowed to remove any terms you want from the sequence? What if you're allowed to rearrange the sequence? What about a series, in which you look at sums of the terms in a sequence?
Lesson  Date  Section  Reading  Assignments  Extras 

12  14Feb  2.1: Metric Spaces (sequences and subsequences) 
pp. 5154  Ch2: 26*  Exams returned and discussed sequences/series 
13  19Feb  2.1: Metric Spaces (metrics; continuity) 
pp. 5156  Ch2: 15,24,82  Miniproject assignments metric spaces 
14  21Feb  2.1: Metric Spaces (open and closed sets) skip Homeomorphism 
pp. 5864  Ch2: 3*,5*,13*  Leonhard_Euler and the Konigsberg Bridges 
15  25Feb  2.1: Metric Spaces (continuity; inheritance) 
pp. 5556,6469  Ch2: 6,17 [Turnin: these problems] 
(Fun Math: donuts & coffee cups) 
16  27Feb  2.1: Metric Spaces (product metrics; Cauchy sequences) 
pp. 7175  Ch2: 27  the taxicab & Facebook metrics 
17  29Feb  2.1: Boundedness  p. 75  Ch2: 29*,30* Paper selection for projects 
contractions & fixed points (Fun Math: The Brouwer fixed point theorem) (Skill: counterexamples) 
18  4Mar  2.2: Compactness I  pp. 76  [Turnin: these problems]  Frechet and the importance of compactness 
19  6Mar  2.2: Compactness II  pp. 7682  Weierstrass; the BolzanoWeierstrass & HeineBorel Theorems  
20  10Mar  2.5: The Cantor Set  pp. 9599  [Turnin: these problems]  Cantor sets 
21  12Mar  2.5: The Cantor Set  pp. 9599  Miniprojects due by 13Mar  fractional dimensions (Fun Math: weird spaces) 
Spring Break 
After Spring break, we will turn to the question: why is compactness SO important? We will also look at the most significant theorems which require the condition of compactness in a very fundamental way. Finally, we'll look at an alternate definition of compactness… which looks very different but turns out to be equivalent.
Lesson  Date  Section  Reading  Assignments  Extras 

22  25Mar  Review of 2.12.2 2.3: Connectedness 
pp. 8287  Bernard_Bolzano the topologist's sine curve 

23  27Mar  Applications of Compactness I  pp. 7982  (Application: optimization)  
24  31Mar  2.4: Coverings  pp. 8895  [Turnin: these problems]  spacefilling curves 
25  2Apr  Applications of Compactness II  (Application: approximations)  
26  4Apr  2.4: Coverings  (Application: sensor coverings)  
27  8Apr  Applications  [First three glossary submissions due]  (Application: voting)  
28  10Apr  Review  [Turnin: these problems as part of WPR grade]  
29  14Apr  WPR II  about WPR II  All glossary entries must be completed by 1600 on 13Apr 
Block III: Functions, Continuity, and Differentiation
In this final block, we will take a look at functions, continuity, and differentiation. Now that the concept of limit has been made rigorous, the concept of a derivative can be as well!
Think About: What conditions must you add to make a continuous function differentiable? What is "lost" (or "gained") at points where a function is not differentiable?
Lesson  Date  Section  Reading  Assignment  Highlights & Extras 

30  16Apr  3.1: Definition of the Derivative Mean Value Theorem 
pp. 139143  
31  18Apr  Course Drop  
32  22Apr  3.1: L'Hopital's Rule and Discontinuities  pp. 143146  [Turnin: these problems]  
33  24Apr  3.1: Power Series and Taylor's Theorem  pp. 149151  
34  28Apr  3.1: Power Series and Taylor's Theorem  pp. 149151  
35  30Apr  Review  
36  2May  WPR III  about WPR III  All glossary entries must be completed by evening on 1May 
Course End
The last few weeks will be spent on review and topics of your choosing. You will get the chance to research, write, and speak about a topic related to analysis.
Lesson  Date  Section  Reading  Assignments  Highlights & Extras 

37  6May  Radius of Convergence/Power Series  3.1,3.3  [Turnin: these problems as part of WPR grade]  
Projects Day (8May)  
38  9May  Project Drop/IPR's  
39  13May  Project Presentations I  
40  15May  Project Presentations II  
TEE  20 May  TEE  Comprehensive  Th319 07351005  Good luck! 