I need to show that
if every subsequence of X = (xn) has a subsequence that converges to 0, then the lim X = 0.
Any help would be greatly appreciated!
I would prove it by showing that a sequence which does not converge to has a subsequence which has no subsequence converging to . This is easy. The sequence must be frequently outside of some -neighbourhood of ; take the subsequence which consists of those points.