Notes on Homework #2
Exercise 1.4.3 We are given real numbers a < b and we have to show that there exists an irrational number 1.4.3 to justify t with a < t < b. Let a = a = 2 and b = b = 2. Since a <
Examples re. Uniform Convergence
These are the examples we discussed in class on Monday, Oct. 26. On the last page you will find our discussion of the uniform convergence of sn on A=[-1,1], and why th
Additional Solutions, in Preparation for Exam #2
Exercise 4.2.3 (a) Consulting Corollary 4.2.5, what we need is two sequences (xn ) and (yn ), both converging to 0, with xn = 0 and yn = 0, and for whi
Notes on Homework #7
Exercise 4.2.1 (c) Most of you had the right idea, writing |x3 8| = |x 2| |x2 + 2x + 4|. Now you need to produce an upper bound for the second factor. If we commit ourselves to ch
Notes on Homework #5
Exercise 2.7.1 c We are going to prove the Alternating Series Test. The hypotheses are that (an ) is a sequence with an 0, a1 a2 an an+1 , and (an ) 0. 1) The important thing to r
Hints for Exercise 2.7.1, Part (c) 1. First show that the partial sums sn are ordered as follows (see the picture on page 35): s2 < s2 < < s2n < < s2n+1 < < s3 < s1 . In other words the even-numbered
Hint for Exercise 2.5.4
This problem requires an argument in several stages. Here is an outline. You need to write it out more completely and supply the explanations needed for each step. 1. The proof