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 definition of what it means for a function f : R R to “be

negative for sufficiently 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 sufficiently 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 differentiable. Justify!

 

 

 

(b) (5 points) Use the definition 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

differentiable everywhere on R. Compute g .

 

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