# Calendar MAT 401 Fall 2013. Jennings.

This is a tentative calendar only. As we go along I'll adjust it as necessary.

• 1st week: Aug. 26, 28
• A little bit of history. Why we need proofs in analysis.
• 0.3 Basic set theory (Review). Cardinality.
• Homework: Exercises 0.3.15, 0.3.17, 0.3.19, 0.3.20. Due Wednesday 9/4. Prove all your results, so for example in exercise 17 prove that you have found all the n's, in exercise 19 explain why the union isn't finite, in exercise 20 explain why the intersection is finite. Don't make your proofs long and complicated unless they have to be.
• 2nd week: Sept. 2, 4
• Labor Day Holiday Monday Sept. 2
• Finish chapter 0
• 1.1 Basic properties of the real numbers. Least upper bound.
• Homework: Exercises 1.1.3, 1.1.4, 1.1.9. Due Wednesday 9/11
• 3rd week: Sept. 9, 11
• 1.2 The set of real numbers, $$\mathbb{R}$$. Archimedian property, sup and inf.
• Homework: Exercise 1.2.4, 1.2.7, 1.2.9, 1.2.10. Due Monday Sept. 23 (a week from next Monday). You'll probably need some help with the last two exercises so we'll work part of them together in class next week.
• 4th week: Sept. 16, 18
• 5th week: Sept. 23, 25
• 1.3 Absolute value.
• 1.4 Intervals and the size of $$\mathbb{R}$$
• 2.1 Sequences and limits.
• Homework: Exercises 1.3.3, 1.3.5. Due next Monday.
• More homework: Exercises 2.1.4, 2.1.5, 2.1.10, 2.1.13. Due next Wednesday.
• 6th week: Sept. 30, Oct. 2
• 2.2 Facts about limits of sequences.
• Test postponed until next Wednesday.
• 7th week: Oct. 7, 9
• 2.2 Facts about limits of sequences.
• Test Wednesday Oct 9 Solutions
• 8th week: Oct. 14, 16
• 9th week: Oct. 21, 23
• Homework: exercise 2.1.6, 2.2.5, also prove that the sequences $$\left\{\frac{n}{2n+3}\right\}_{n=1}^{\infty}$$ and $$\left\{\frac{n}{2n-3}\right\}_{n=1}^{\infty}$$ converge. For the proof use the definition of limits, so it should be an $$\epsilon$$-type proof. Due Monday Oct. 28.
• 10th week: Oct. 28, 30
• 2.3 Limit superior, limit inferior, and the Bolzano-Weierstrass Theorem.
• 2.4 Cauchy sequences.
• Homework exercises 2.3.1, 2.3.2, 2.3.5, 2.3.6, 2.3.7. Due next week.
• 11th week: Nov. 4, 6
• 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.
• 3.2 Continuous functions.
• 12th week: Nov. 11, 13
• Veterans' Day Holiday Monday Nov. 11
• 3.3 Extreme value theorem (the text calls this "Min-max theorem") and intermediate value theorem.
• 13th week: Nov. 18, 20
• Test Wednesday Nov. 21 Solutions
• skip 3.4 Uniform continuity for now
• 4.1 The derivative
• 14th week: Nov. 25, 27
• 4.2 Mean value theorem
• 4.3 skip Taylor's theorem
• 15th week: Dec. 2, 4
• Final Exams week:
• Final Exam Wednesday Dec. 11 5:30 - 7:30pm