1) It is well known that between a prime $p$ and $p^2$ always exist a prime, but what is the shortest proof of that (by elementary methods or not)?

(One can say that we can have it as a collorary of Bertrand's postulate, but it is a stronger result.)

2) I ask it as an example, are there any characteristic examples of results that their first proof was really large comparing to some proof that someone found later?

3) Or of results that their only known proof/proofs is/are a collorary of the proof of something stronger? (Maybe the one that I give is not an example for this.)

NOTE:I was asked to change the title to a more precise

isa good example of what you're asking for: off the top of my head, I don't know a shorter proof than the one which establishes Bertrand's Postulate (i.e., an elementary but somewhat tricky couple of pages), which is a much stronger result. But I think your question "[A]re there any results that their first proof was large comparing to some proof that someone found later?" is far too broad for this site. (Certainly the answer is yes: a large percentage of first proofs are longer than what is eventually found.) – Pete L. Clark Jan 14 '11 at 11:22