Definition of convergence. In words: Sequence covnerges to point if for every neighbourhood of the point, there is a finite point in the sequence such that all points forward from that point in the sequence are in that neighbourhood.

In a Hausdorff space if a sequence converges, its limit point is unique (proof)

Another proposition saying that a Limit point of a set in a Hausdorff space intersects the set in infinitely many points. Another proof. We need stronger condition to say that a limit point is the point to which a sequence converges.