Writing Intensive Homework II

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The second writing intensive assignment will be submitted twice, first as a rough draft and second as the final draft. The rough draft is due October 31st, the final draft due one week after the rough draft is returned. The rough draft will probably be returned on November 12th. When you're handing in the final draft, attach the rough draft also. Failing to do so will result in loss of substantial part of the points.

The assignment has four parts:

part a) Give a careful exposition of the following four concepts: function, sequence, limit of a function, limit of a sequence.
part b) Let f be a function defined on an interval (a,b) of the real line. Let , $n=1, 2, \ldots,$ be a sequence of real numbers in (a,b) such that $s_n \to a$ . Suppose that

$\displaystyle \lim_{x\to a^+} f(x) = L \;.$
Give a complete proof that

$\displaystyle \lim_{n \to \infty} f(s_n) = L \;.$
part c) Using part b), prove that

$\displaystyle \lim_{s\to O^+} \sin\left( \frac{1}{x} \right)$
does not exist.
part d) Let g be a function defined on an interval (u,v) of the real line, and let K be a real number. Suppose that for every sequence , $n=1, 2, \ldots,$ of real numbers in (u,v) such that $t_n \to u$ , we have $g(t_n)\to K$ . Give a complete proof that

$\displaystyle \lim_{x\to u^+} g(x) = K\; .$

Good luck!

Reference materials: The first two of the following books are at the Mathematics Library reference, you may use them only in the library. The last one is on the shelf, please try to use it in the library and do not check it out unless it is absolutely necessary to do so.

Higham, Nicholas J., Handbook of Writing for the Mathematical Sciences, Society for Industrial and Applied Mathematics, 1993.
Gillman, Leonard, Writing Mathematics Well : A Manual for Authors, Mathematical Association of America, 1987.
Knuth, Donald E., Larrabee Tracy, and Roberts, Paul M., Mathematical Writing, Mathematical Association of America, 1989.

Please check the checklist page for some comments on how to type and organize your paper.

Good luck!