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 < b we can apply Theorem saying that there is a rational
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 the convergence of tn is not uniform.
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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 which lim |xn | and lim |yn | disagree. For these we can t
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 choosing 1 then |x 2| < implies that |x 2| < 1, so that 1
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 recognize is that the even-numbered partial sums end wit
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 partial sums form an increasing sequence, the odd-numbe
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 will be by contradiction. Suppose (an ) does not conve