Limits of sequences are unique when they exist, as distinct points are separated by some positive distance, so for \ epsilon less than half this distance, sequence terms cannot be within a distance \ epsilon of both points.
12.
A first countable, separable Hausdorff space ( in particular, a separable metric space ) has at most the closure is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of X.
