Last updated Fri Jan 13 15:59 EST 2017
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Due-date: Wednesday, 1/18/17
You are required to do all of the problems below. You will not be required to hand them all in. I have indicated below which ones you do have to hand in on the due-date. Don't make the mistake of thinking that I'm collecting only the problems I think are important.
The "due date" above is the date that your written-up problems should be handed in, but don't wait to get started on the assignment. You should always get started on problems as soon as we cover the relevant material in class.
- A. Rosenlicht p. 94/ 32, 34, 35, 36 (prove your answer), 37, 38. Of these, hand in only 32, 34ac (i.e. the first and third sequences in the problem), 38 (modified as indicated below). Warning: for sequences of functions, Rosenlicht's "converges", absent the modifier "uniformly", is my "converges pointwise" (see p. 83).
In #32, better wording to indicate the sequence of functions would be "the sequence of functions \( (f_n:[0,\infty)\to{\bf R})_{n=1}^\infty\) defined by \( f_1(x)=\sqrt{x},\ f_2(x)=\sqrt{x+\sqrt{x}},\ f_3(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}, \dots\)". A similar comment applies to problems 34–36.
Note: #32 is tricky and (somewhat) hard. It is relatively easy to figure out, for each \(x\geq 0\), the only possible number to which the sequence \(f_n(x)\) could converge, where \(f_n(x)\) is the \(n^{\rm th}\) term of the sequence indicated in the book. The tricky part is that you are likely to (initially) find two possible values, and discard one of them as being impossible, but the one you discard will actually be the right answer for one value of \(x\). The hardest part of the problem, once you figure out the only possible limit-function \(f\), is proving that the sequence actually does converge (pointwise) to \(f\). Remember that you can't prove that something is true by assuming it's true.
In #38, for any set \(X\) and metric space \(Y\), a function \(f:X\to Y\) is bounded if the range of \(f\) is a bounded subset of \(f\). In this problem, assume only that the domain \(X\) of the functions in your sequence is a nonempty set; it is irrelevant whether \(X\) is a metric space.
- B. Click here for non-book problems. Of these, hand in only B1bc and B2b. For B1b, hand in only the proof of the triangle inequality. In doing parts of B1 and B2, you are allowed to assume the results of earlier problem-parts on this page, whether or not you were successful in proving them.
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