Calendar for MAT 401 Advanced Analysis, Fall 2012

I will post homework assignments here so please check back often.

Section numbers refer to our textbook Basic Analysis: Introduction to Real Analysis by Jiří Lebl.

• 1st week: Aug. 27, 29
  • A little bit of history. Why we need proofs in analysis.
  • 0.2 About analysis. (Read on your own)
  • 0.3 Basic set theory (Review). Cardinality.
  • Homework: Exercises 0.3.5 (tricky!), 0.3.9d, 0.3.11, 0.3.17 (prove that your answer is true), 0.3.19, 0.3.20. Due Monday Sept. 3. (There was a typo on my original version of this assignment; it said 0.3.12 when it should have been 0.3.17).
• 2nd week: Sept. 3, 5
  • Labor Day Holiday Monday Sept. 3
  • Finish chapter 0
• 3rd week: Sept. 10, 12
  • 1.1 Basic properties of the real numbers. Least upper bound.
  • Homework: 1.1.3, 1.1.6, 1.1.9. Due next Monday.
  • 1.2 The set of real numbers, \(\mathbb{R}\). Archimedian property, sup and inf.
  • Homework: 1.2.1, 1.2.2, 1.2.7 (hint: is \((\sqrt{x}-\sqrt{y})^2\) positive, or negative, or what?), (1.2.9 was originally part of this but I postponed it until next week.). Due next Wednesday.
• 4th week: Sept. 17, 19
  • Homework: Prove the sup part of exercises 1.2.9 and 1.2.10. (Hint: use prop. 1.2.8). Due Monday.
• 5th week: Sept. 24, 26
  • 1.3 Absolute value. Homework: 1.3.2a, 1.3.3, 1.3.5. Due next Monday.
  • 1.4 Intervals and the size of \(\mathbb{R}\)
  • 2.1 Sequences and limits. Homework: 2.1.6, 2.1.10, 2.1.12, 2.1.13, 2.1.16. Due next Monday.
• 6th week: Oct. 1, 3
  • 2.2 Facts about limits of sequences. Homework: exercises 2.2.3, 2.2.5, 2.2.7 (hint: what if the signs of the \(x_n\)s flop back and forth?), 2.2.8. Due next Monday.
  • Test Wednesday Oct. 3 Solutions
• 7th week: Oct. 8, 10
  • 2.2 Facts about limits of sequences (continued). Homework: exercises 2.2.3, 2.2.5, 2.2.7 (hint: what if the signs of the \(x_n\)s flop back and forth?), 2.2.8. Due next Monday.
• 8th week: Oct. 15, 17
  • Discuss answers to test questions.
  • 2.3 Limit superior, limit inferior, and the Bolzano-Weierstrass Theorem. Homework: 2.3.1, 2.3.2, 2.3.5, 2.3.6. Due next Wednesday.
• 9th week: Oct. 22, 24
  • Finish 2.3: Bolzano Weierstrass theorem. (See homework last week).
  • 2.4 Cauchy sequences. Homework: exercises 2.4.1, 2.4.2, 2.4.5. Due next Wednesday. Due Wednesday Oct. 31.
• 10th week: Oct. 29, 31
  • 2.5 Series: We'll only touch on this very briefly; it was covered pretty well in your calculus class. I think the calculus proofs using the integral test are simpler and easier to rememeber than the ones in our text, but of course we haven't developed integration yet so they wouldn't work in this course.
  • 3.1 Limits of functions. This introduces the famous \(\epsilon-\delta\) definition, which is an extension of the ideas we have already discussed. Homework (corrected): exercises 3.1.4 parts i) and iii), 3.1.7, 3.1.8, 3.1.10 The previous "hint" for problem 3.1.10 was too hard. Here is an easier approach: The hypotheses of the problem give you a lot of Cauchy sequences \(\{f(x_n)\}_{n=1}^{\infty}\). If these Cauchy sequences all converge to the same converge to the same limit \(L\) then you can use lemma 3.1.7 to conclude that \(f\) is continuous at \(c\). So the only thing that could go wrong is that some of these Cauchy sequences might converge to different limits. Let's show that can't happen. If it does happen then you'll have two sequences \(\{x_n\}_{n=1}^{\infty}\) and \(\{x'_n\}_{n=1}^{\infty}\) that converge to \(c\), but \(\lim_{n\rightarrow\infty} f(x_n) = L\) and \(\lim_{n\rightarrow\infty} f(x'_n)=L'\) where \(L\neq L'\) are different limits. Show that you could put these two sequences together to construct a sequence that tries to approaches both limits in the same way that the sequence \((-1)^n\) tries to approach both \(1\) and \(-1\). Such a sequence can't be Cauchy, which contradicts the hypotheses of this problem.
• 11th week: Nov. 5, 7
  • Test postponed until Monday Nov. 19
  • 3.2 Continuous functions. Homework: 3.2.1, 3.2.2, 3.2.3, 3.2.10, 3.2.11. For full credit work 3.2.1, 3.2.2, and 3.2.3 directly, using the $\epsilon-\delta$ definition of continuity (Def. 3.2.1) instead of using sequences or lemma 3.1.7. Example 3.1.5 shows how to do this in one example. I will give you another example next Wednesday
• 12th week: Nov. 12, 14
  • Veterans' Day Holiday: Monday Nov. 12
  • 3.3 Extreme value theorem (the text calls this "Min-max theorem") and intermediate value theorem. Homework 3.3.1, 3.3.2, 3.3.6, 3.3.8
• 13th week: Nov. 19, 21
• Test Monday Nov. 19 Solutions
• skip 3.4 Uniform continuity for now
• 4.1 The derivative
• 4.2 Mean value theorem
• 4.3 skip Taylor's theorem
• Thanksgiving Holiday: Thursday Nov. 22 - Friday Nov. 23
• 14th week: Nov. 26, 28
  • Homework: exercise 4.1.1. Due Wednesday Dec. 5
• 15 th week: Dec. 3, 5
  • 5.1 The Riemann integral
  • 5.2 Properties of the integral
  • 5.3 Fundamental theorem of calculus
• Final exams week: Dec. 10-14
  • Final exam: Wednesday Dec. 12 5:30-7:30pm Solutions