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Differentiability, Bounded Above and Supremums

1. Let A and B be two nonempty sets of real numbers. Define A+B = {a+b: a belongs to A and b belongs to B}. (a) Show that if A is open, then A+B is open. (b) If A and B are both closed, is A+B closed? Justify your answer.

2. Let f be differentiable for x > a and A as x --> infinity. Prove that there is a sequence x_n --> infinity such that f'(x_n) --> A. Give an example to show that f'(x) may not tend to A as x --> infinity.

3. Let f be differentiable for x > a and f(x) + f'(x) --> A as x --> infinity. Prove that f(x) --> A and f'(x) --> 0.

4. If A and B are nonempty subsets of R that are bounded from above, prove that the sup(A+B) = sup(A)+sup(B)

5. Let the points of any countable subset E of (a,b), which may be dense, be arranged in a sequence {x_n}. Let {c_n} be a sequence of positive numbers such that converges. Define f(x) = SUM (a < x <b) (i.e., sum over those indices n for which x_n < x). Verify the following properties of f:
(a) monotonically increasing on (a,b);
(b) discontinuous at every point of E; in fact, .
(c) continuous at every other point of (a,b).

6. Let A = {(x, y) 0 <= x, y <= 1}. If f is a continuous function from A to R, can f be one-to-one? Justify your answer.

7. If SUM a_n converges and if {b_n} is monotone and bounded, prove that SUM a_n b_n converges.

8. Prove that f(x) = x^1/2 is uniformly continuous on [0,infinity).

Solution Summary

This solution provides step by step proof for each part of the question in an multi-paged attached .rtf file.