How do you prove that if a sequence is bounded then it has only one convegent subsequence?
Monotonicity was not part of the original problem statement, however. Also, if a sequence converges (as is the case if it is bounded and monotone), then every of its subsequences converges to the same limit (I don't even consider finite subsequences here, since talk of convergence makes hardly any sense for finite sequences).
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.
- Hollywood