Homework for Math 103
Assignment Four
Due 6 December 1999
Work problems 2, 3, 4, 5, 13, 16, 18, 20 on pages 227-230 of your text (end of chapter ten).

You will want to consider the hints below. In addition, I found the following useful on problem 16. Recall that if then .

HINT FOR #2: Either use the Baire Category Theorem or the observation that if is not a polynomial, then given , , and , there is a such that . HINT FOR #4: Estimate . HINT FOR #5: The hypotheses of the problem don't allow us to conclude even that the limit function is continuous. Instead, you'll have to prove that is uniformly Cauchy. You may want to use (and prove) that if for all and converges pointwise to a 0, then

HINT FOR #13: Use the residue theorem and #8. HINT FOR #18: This is a straightforward modification of the argument used in class for the case . HINT FOR #20: Notice that uniformly on compact subsets of . Show this implies uniformly on any provided on .