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 $\displaystyle \{x_n\}$ which does not converge to $\displaystyle 0$ has a subsequence which has no subsequence converging to $\displaystyle 0$. This is easy. The sequence $\displaystyle \{x_n\}$ must be frequently outside of some $\displaystyle \epsilon$-neighbourhood of $\displaystyle 0$; take the subsequence which consists of those points.