Prove the convergence of a sequence based on sub-sequence Suppose that every convergent sub-sequence of $a_n$ converges to zero. Prove that $a_n$ converges to zero. I tried to prove this statement using the definition of limit. From the question, I know that $a_{n_k}$ converges to zero $\forall$ k $\in$ N. I want to show that $\forall \epsilon > 0, \exists N \in$ natural number such that $\forall n > N$, we have |$a_n - 0$| $< \epsilon$

First of all, $(a_{n})$ must be bounded, if not, one can find some $|a_{n_{k}}|$ such that $|a_{n_{k}}|\rightarrow\infty$, if your convergence includes the sort of infinity, then this violates the assumption.

Now let a subsequence $(a_{n_{k}})$ such that $\lim_{k\rightarrow\infty}|a_{n_{k}}|=\limsup_{n\rightarrow\infty}|a_{n}|$, so by assumption $\limsup_{n\rightarrow\infty}|a_{n}|=0$. Hence $0\leq\liminf_{n\rightarrow\infty}|a_{n}|\leq\limsup_{n\rightarrow\infty}|a_{n}|=0$ and we have then $\lim_{n\rightarrow\infty}|a_{n}|=0$.