Past analysis qualifying exam part 2

I plan to take the analysis qualifying exam next time it is offered and I would like detailed solutions to a previous exam (some of the problems are fairly basic, but I would like to see solutions to all eight problems to be thorough). Note: This is for problems 3 and 4. (See attached file for full problem description with proper symbols) --- Analysis Qualifying Exam Spring 2004 1. Let A and B be two nonempty sets of real numbers. Define A+B = {a+b: a A and b 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 be differentiable for and A as . Prove that there is a sequence such that A. Gi... click for more

Past analysis qualifying exam part 3

I plan to take the analysis qualifying exam next time it is offered and I would like detailed solutions to a previous exam (some of the problems are fairly basic, but I would like to see solutions to all eight problems to be thorough). Note: This is for problems 5 and 6. (See attached file for full problem description with proper symbols) --- Analysis Qualifying Exam Spring 2004 1. Let A and B be two nonempty sets of real numbers. Define A+B = {a+b: a A and b 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 be differentiable for and A as . Prove that there is a sequence such that A. Gi... click for more

Past analysis qualifying exam part 4

I plan to take the analysis qualifying exam next time it is offered and I would like detailed solutions to a previous exam (some of the problems are fairly basic, but I would like to see solutions to all eight problems to be thorough). Note: This is for problems 7 and 8. (See attached file for full problem description with proper symbols) --- Analysis Qualifying Exam Spring 2004 1. Let A and B be two nonempty sets of real numbers. Define A+B = {a+b: a A and b 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 be differentiable for and A as . Prove that there is a sequence such that A. Gi... click for more

Problem

(See attached file for full problem description) I am using the book Methods of Real analysis by Richard Goldberg. --- Let and let D be a dense subset of E. If are continuous real-valued functions on E for n=1,2,…, and converges uniformly on D, prove that converges uniformly on E. ---

Proof

If the conjecture is true, prove it. If it's false, prove that its false by counterexample or a proof by contradiction. 2. Denote by -P the set of all negative integers, i.e., the set to which the number m belongs only in case there is a member n of P such that m = -n. If the number m is in -P and Z is a number such that m