Real Analysis.....im sup Inequalities

Prove if... and... are bounded sequences of real numbers, then lim sup... (See attachment for full question)

Real Analysis...Open Sets

Prove that the open interval...with a, b being real numbers is an open set. (See attachment for full question)

Real analysis, Convergent sequences and Limits.

Prove the sequence defined by a1 = 0, a2=1 and a(n+2)= (a(n+1)+a(n))/2 for n>=0, converges and find the limit. (See attachment for full question)

One Dimensional Normed Linear Space

Suppose that E is a one-dimensional normed linear space. Prove that E is complete and that each linear functional on E is continous.

Functional Analysis - Continuity, Graphing

Problem: Let f be that function defined by setting (PLEASE SEE ATTACHMENT FOR SETTING) a. Describe graphically f(x). b. At what points is f continuous? (This is a graduate course in real analysis. It seems that so many have attempted to solve this question but to no avail! Please give it a try).