prove that $\lim \limits_{n \to \infty}a_n= \lim \limits_{n \to \infty}b_n$

$\{a_n\}$ and $\{b_n\}$ are two converging sequences.

Its given that the two sets : $\lbrace n \in \mathbb{N} : a_n \le b_n \rbrace$ , $\lbrace n \in \mathbb{N} : b_n \le a_n\rbrace$ are not bounded.

prove that $\lim \limits_{n \to \infty}a_n= \lim \limits_{n \to \infty}b_n$

any hints? (intending to formally prove this)

Hint: consider the sequence $a_n - b_n$, which we know must converge. Suppose that $\lim_{n \to \infty}(a_n - b_n) \neq 0$, and derive a contradiction.
If a sequence converges, then every subsequence converges to the same limit. In particular, you have a sequence $n'_k$ and a sequence $n''_k$ of integers such that $a_{n'_k} \leq b_{n'_k}$ and $a_{n''_k} \geq b_{n''_k}$. Taking limits as $k \to +\infty$, and recalling that these limits coincide with the limits of the whole sequences, $\lim_n a_n \leq \lim_n b_n$ and $\lim_n a_n \geq \lim_n b_n$. Hence the conclusion follows.