I know there are proofs of this statement on the website but I was wondering if my, seemingly too simple proof is valid or not.
Assume that $\lim \inf a_n=\lim \sup a_n=x$
We know that $\inf_{k\leq n} a_k\leq a_n\leq\sup_{k\leq n}a_k$ which by squeeze theorem means $a_n$ converges to x. Is this correct?
This seems way too easy, all other proofs I have seen seem to use the definition.