Formal definition of limits and proofs using them.

I have a couple of problems due soon that I need help comprehending.

Firstly, I need to prove the Squeeze Theorem using the formal (Epsilon-Delta) definition of limits. I've read the actual proof from Wikipedia, but I just can't wrap my head around it, it feels like they're skipping some sort've explination in certain parts.

For example, on the Wiki page (Squeeze theorem - Wikipedia, the free encyclopedia) it gets to a point where they give http://upload.wikimedia.org/math/3/d...0b53c4e829.png, then jump down and say that this proves http://upload.wikimedia.org/math/c/9...9981ab2106.png, but I don't see how they got from one step to the other.

Then, while they prove the general case, they have http://upload.wikimedia.org/math/0/d...2a89301554.png and end up with http://upload.wikimedia.org/math/f/7...42978c53f5.png. I understand the two steps they show in between, but I don't see how using those two steps results in the latter equation.

Secondly, I need to prove that: lim x->infinity f(x) is equal to lim x->0+ f(1/x), not intuitively, but using:

The definition of lim x-> infinity f(x) = L is: for all epsilon > 0 there exists an N > 0 such that x > N -> |f(x)-L| < epsilon.

Preemptive thanks for the help!