Given a convergent sequence $\{x_n\}$ prove that either: $$ \lim_{n \to\infty}\{x_n\} = \lim_{n \to \infty} \sup \{x_n\} $$ or $$ \lim_{n \to\infty}\{x_n\} = \lim_{n \to \infty} \inf \{x_n\} $$
I believe this problem has been solved several times here, but i couldn't find such a question (probably due to translation issues, since the original problem is in another other).
I've started with gathering what is given in the problem statement. So we have that a sequence is convergent, thus: $$ \lim_{n\to\infty}x_n = L \iff \{ \forall\varepsilon >0, \exists N\in \mathbb N:\forall n> N \implies |x_n-L|<\varepsilon \} $$
Also we have that the sequence is bounded, so: $$ m = \inf\{x_n\} \le x_n\le \sup\{x_n\} = M \\ m \le x_n \le M $$
Now using these facts I believe I should make some assumption (for example that $x_n$ doesn't reach any bound and proceed by contradiction), but i can't wrap my mind for several hours already.
I would appreciate if someone could show me how to prove this or point to an already answered question.