Bolzano-Weierstrass Theorem

Somewhat intuitive, but not without some technical merit, this is an important result in Analysis: Every bounded real sequence has a convergent subsequence. The proof is fairly clever. You keep chopping the bound in half and taking whichever half has an infinite number of elements remaining in the sequence (if both, just pick one). The endpoints of these increasing tight bounds converge and, by the squeeze theorem, the subsequence of elements that resides in each also converges.