Convergence of sequence with subsequences Let ${a_n}$  be a bounded sequence such that any of its convergent proper subsequences converge to the same limit $l$. Prove that ${a_n}$ converges to $l$.

Any hints please?
Hint: Suppose $a_n$ does not converge to $l$. Then there is $\varepsilon \gt 0$ and an increasing sequence $n_k \in \mathbb{N}$ such that $|a_{n_k} - l| \gt \varepsilon$ for any $k \in \mathbb{N}$. By Bolzano-Weierstrass $a_{n_k}$ has a convergent subsequence. Show that the limit of this subsequence cannot be $l$ and use this to conclude.
It would be a good start assuming that the sequence does not converge to $l$.
Then try to find a proper subsequence that does not converge to $l$.