$\liminf a_n\leq \lim a_{n_{k}} \leq \limsup a_n$

Question

So we need to prove that the limit of any subsequence is sandwiched between the $\limsup$ and $\liminf$ of the original sequence.

The professor started off by showing that there is always a subsequence that converges to $\limsup a_n$, and also there is always a subsequence that converges to $\liminf a_n$. He also showed that $\limsup a_{n_{k}}$ is less than or equal to $\limsup a_n$. This part I also understand. But then how do you use these two facts to conclude that $\lim a_{n_{k}} \leq \limsup a_n$?

Answer 1

$\lim a_{n_k} \le \limsup a_{n_k}$!