Let a sequence $(t_{n})_{n\in\mathbb{N}}$ in $\mathbb{R}$ and for any subsequence $(t_k)_{k\in\mathbb{N}}\subset(t_{n})_{n\in\mathbb{N}}$ there exists another subsequence $(t_l)_{k\in\mathbb{N}}\subset(t_{k})_{n\in\mathbb{N}}$s.t. $t_{l}\to t$ as $l\to\infty$. Then, $t_{n}\to t$ as $n\to\infty$.
I know how to prove this using contradiction argument but I am considering a direct proof for this argument. Is there any hint to prove the claim above without contradiction?
Any help is much appreciated!