How do you prove that if a sequence is bounded then it has only one convegent subsequence?
And just to mention another possible criticism of the original question: if a sequence has a convergent subsequence, it necessarily has infinitely many (but, of course, it might be that they all converge to the same limit). And of course, if a sequence converges, all of its subsequences converge, and converge to the same limit at that.
I think we're looking for this:
Bolzano-Weierstrass theorem - Wikipedia, the free encyclopedia
Any bounded sequence has a convergent subsequence. But not necessarily only one, as Failure's counter-example shows.