Another good example is the Johnson-Lindenstrauss Lemma that says that any $n$ points in a Hilbert space can be embedded in a $O(\log n)$-dimensional Euclidean space with distances preserved upto any factor. It turns out that JL-style results crop up in many different versions, the main result itself has proofs ranging from 1 page to 10 pages, and it just keeps on giving :)