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Math 409
Spring 2016
Journal
 Thursday, May 5
 The final examination was given, and solutions are available.
 Thursday, April 28
 We reviewed for the final examination, which takes place from 3:00 to 5:00 on the afternoon of Thursday, May 5.
 Tuesday, April 26
 We discussed properties of the set of Riemannintegrable functions and the Kurzweil–Henstock generalization of the Riemann integral.
 The assignment for next time is to make a list of the three most important topics from each chapter that we covered.
 Thursday, April 21
 We discussed Lebesgue’s theorem stating that a bounded function is Riemann integrable if and only if the set of discontinuities of the function is a set of measure zero. Also we discussed the notion of measure zero and the construction of the Cantor set, an uncountable set of measure zero.
 The assignment for next time (not to hand in) is Exercises 2.13.11, 4.6.9, 5.9.15, and 7.9.6.
 Tuesday, April 19
 We discussed properties of the integral as defined by Cauchy and the notion of convergence of improper integrals.
 Thursday, April 14
 We proved Cauchy’s theorem about the existence of the integral of a continuous function.
 Here is the assignment to hand in on Thursday, April 21: Exercises 8.2.2, 8.2.5, 8.2.9, 8.3.2, 8.4.4, and 8.5.10.
 Wednesday, April 13
 I updated the solutions to the second exam by adding a third solution to problem 5a due to one of you.
 Tuesday, April 12
 The second exam was given, and solutions are available.
 Thursday, April 7
 We looked at some examples of limits for which l’Hôpital’s rule is not a useful tool, we applied the meanvalue theorem to prove the quadratic version of Taylor’s theorem and stated the general version, and we looked at some standard Taylor series.
 Tuesday, April 5
 We proved that a twicedifferentiable function with weakly positive second derivative satisfies the midpoint convexity inequality; we deduced from the concavity of the natural logarithm function that the arithmetic average of positive numbers exceeds the geometric average; and we applied Cauchy’s meanvalue theorem to prove the basic version of l’Hôpital’s rule.
 The assignment for Thursday, April 7 (not to hand in) is to make lists of the major concepts and the major theorems covered in Chapters 5 and 7.
 Thursday, March 31
 We discussed Dini derivates, Darboux’s theorem stating that the derivative of a differentiable function has the intermediatevalue property, and the general notion of convexity based on inequalities.
 Tuesday, March 29
 In class, we discussed meanvalue theorems and some applications.
 Here is the assignment to hand in on Tuesday, April 5: Exercises 7.5.6, 7.6.9, 7.6.14, 7.7.5, 7.8.2, and 7.9.5.
 Thursday, March 24
 We discussed the chain rule (the fake proof and a real proof using the definition of differentiability based on reduction to continuous functions), Exercise 7.3.16, and examples of functions like \(x^2 \sin(1/x)\) having discontinuous derivatives.
 Tuesday, March 22
 In class, we looked at three definitions of differentiability and applied them to solve Exercise 7.2.3.
 The assignment to hand in on Tuesday, March 29 is Exercises 7.2.6, 7.2.12, 7.2.16, 7.2.26, 7.3.10, and 7.3.14.
 Thursday, March 10
 We discussed some applications of the big theorems from last time: namely, a continuous function maps a compact interval to another compact interval, and a continuous function that maps a compact interval into itself has a fixed point (at least one). Additionally, we discussed the concept of uniform continuity, and we saw that a function having a bounded derivative is necessarily a Lipschitz function and therefore is uniformly continuous.
 Enjoy Spring Break!
 Tuesday, March 8
 We discussed two big theorems about continuous functions: namely, a continuous function with a compact domain attains extreme values, and a continuous function whose domain is an interval has the intermediatevalue property.
 Thursday, March 3
 In class, we discussed various equivalent ways to formulate the notion of continuity of a function.
 The assignment to hand in on Thursday, March 10 is
Exercises 5.2.40, 5.3.7, 5.4.4, 5.4.20, 5.4.21, and Challenging Problem 5.10.1.
 Tuesday, March 1
 We continued the discussion of limits of functions and looked at some examples, including the ruler function.
 Thursday, February 25
 We discussed three equivalent ways to understand the notion of limit of a function.
 The assignment to hand in on Thursday, March 3 is
Exercises 5.1.10, 5.1.26, 5.1.36, 5.2.12, 5.2.17, and 5.2.37.
 Tuesday, February 23
 The first examination was given, and solutions are available.
 Next class, we will start Chapter 5 about continuous functions.
 Thursday, February 18
 We reviewed for the examination to be given on Tuesday, February 23, on Chapters 1, 2, and 4. We also solved in groups Exercises 2.11.6 and 2.11.7 about convergent sequences.

In view of the upcoming exam, there is no assignment to hand in next week.
 Tuesday, February 16
 Reminder: Exam 1 on Chapters 1, 2, and 4 takes place in class next Tuesday, February 23.
 In class today, we discussed several ways to characterize compact subsets of the real numbers: every sequence in the set has a subsequence that converges to a point of the set; the set is closed and bounded; the set satisfies the Heine–Borel covering property (“every open cover has a finite subcover”).
 The assignment for Thursday, February 18 (not to hand in) is to make lists of (a) the major concepts and (b) the major theorems covered in Chapters 1, 2, and 4.
 Thursday, February 11
 In class, we discussed the notions of open sets and closed sets.
 The group assignment to hand in on Tuesday, February 16 is Exercises 4.2.24, 4.2.25, and 4.3.9. Please turn in one paper per group.
 Tuesday, February 9
 We discussed Cauchy’s criterion for convergence and various terms used to describe sets of real numbers: neighborhood, interior point, boundary point, accumulation point, isolated point. And we solved Exercise 4.2.1 in groups.
 Thursday, February 4
 In class, we discussed countable sets, Cantor’s theorem about uncountability of the real numbers, and the notion of limit superior.

Here is the assignment to hand in on Thursday, February 11.
 Exercises 2.3.4, 2.9.4, 2.11.3, 2.12.7, 2.13.9, and
 Challenging Problem 2.14.12. Notice in the hint that the first inequality is intended to be
\[
\lefty(n+2m\pi)\right \lt \varepsilon
\]
with a pair of parentheses.
 Tuesday, February 2
 We discussed the squeeze theorem (sandwich theorem), the notion of a subsequence of a sequence, and the Bolzano–Weierstrass theorem. We constructed an example of a sequence with the property that for every real number between 0 and 1 (inclusive), there is a subsequence converging to that number, but there is no subsequence converging to any other real number.
 Thursday, January 28
 In class, we discussed the interaction between limits and the completeness property of the real numbers (namely, the theorem that a bounded monotonic sequence converges) and the interaction between limits and the order relation on the real numbers.
 Here is the assignment to hand in on Thursday, February 4.
 Exercises 2.4.15, 2.5.6, 2.8.9,
2.9.2, 2.11.11, and
 Challenging Problem 2.14.2. To prove the statement that
\[\lim_{n\to\infty} a_n = \lim_{n\to\infty} s_n
\]
requires showing two things: both that convergence of the sequence \(\{a_n\}\) implies convergence of the sequence \(\{s_n\}\) and that the two limits are equal.
 Tuesday, January 26
 We discussed the formal definition, due to Cauchy, of convergence of a sequence.
 Thursday, January 21
 In class, we discussed the completeness axiom for the real numbers, induction, the Archimedean principle, and the density of the rational numbers in the real numbers.
 Here is the assignment to hand in on Thursday, January 28.
 Exercises 1.3.5, 1.4.1, 1.6.22, 1.7.4, 1.9.8,
and
 Challenging Problem 1.11.6.
 Tuesday, January 19
 At the first class meeting, we discussed the meaning of \(\R\) (the set of real numbers) being an ordered field. The additional concept of completeness is the first item of business for the next class meeting.
The assignment is to read in Chapter 1 of the textbook. What we have covered so far is Sections 1.1–1.4.