Let $\{a_n\}_{n=1}^{\infty}$ be a sequence such that every subsequence has a subsequence that converges to $L$. Then, $\lim_{n \to \infty} a_n = L$.
If a sequence $(u_n)$ be such that its every subsequence has a subsequence that converges to $0$, then $\lim u_n= 0$