# Project #48027 - advanced math problem ( Introduction-Real Analysis)

I need it within 15 hours

 1. (10 points) Let L ∈R and let {an }n be a sequence of real numbers such that all its monotone subsequences converge to L. Prove that {an }n converges to L. Is the converse of this statement true (formulate and prove it if so)?

 2. (a) (3 points) Write a mathematical deï¬nition of what it means for a function f : R → R to “be negative for suï¬ƒciently large |x|”.

 (b) (7 points) Let f : R → R be a continuous function such that f (0) = 1 and such that f (x) is negative for suï¬ƒciently large |x|. Prove that there exist at least two distinct real numbers a such

that f (a) = a2 .

3. (a) (8 points) Prove that if f, g : R R are bounded and uniformly continuous, then the

product function f g is also uniformly continuous on R.

 (b) (2 points) Is the above statement still true if we drop the boundedness assumption?

4. (a) (2 points) Find the points (if any) where the function f (x) = |x| is not diï¬€erentiable. Justify!

(b) (5 points) Use the deï¬nition of derivative to compute f (x) if f (x) = x|x| for all x R.

 (c) (3 points) Use (b) and the product rule to show that if n ∈N, then the function g(x) = xn |x| is diï¬€erentiable everywhere on R. Compute g .

 Subject Mathematics Due By (Pacific Time) 11/18/2014 12:00 am
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