I think you must have missed something;I assume your sequence is bounded and monotone,but still this has infinite convergent sub-sequences converging to one limit.

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.