Advanced Calculus - Convergence Sequence Problem

I had been trying to solve these problems for the entire weekend, and I just can't think of anything that would work, please help!!!

#1. Let c be a real number. Prove that there is an n of Natural Number such that 2^n > c

My solution: 2^n is not bounded so I can't use that to prove c can be a lower bound, then I tried to use the archmedian principle and it still won't help.

#2. Let S = (k/2^n : k within Natural Number, n without Natural Number). Let 0 <= a < b. Prove that there is an x within S such that a < x < b.

My solution: I think this is to prove that S is dense, but the 2^n is really messing up my pace here, I can't prove this problem as like other "dense" set...

#3. Let a1 = 4. Define an+1 = an/2 + 2/an , for n = 1,2,3,...

a) Evaluate a2, a3, a4.

b) Prove that if an > 2, then an+1 > 2. use induction to prove that an > 2 for all n within the set of Natural Numbers.

c) Prove that an+1 <= an for all n within the set of Natural Numbers.

d) Is {an} bounded? If so, find A,B of R such that A <= an <= B for all n within N.

e) Does lim (n->inf) an exist? Why?

My solution: I'm currently still reading the text in attempt to find the answer.

Finally, I understand that these problems are hard and would reqire some work to finish, but if you can as much as give me some hints I would really appreicate it!!!

Thank you!