Show that lim inf $a_{n}$ $\leq$ lim sup $a_{n}$

Question

Show that lim inf $\{a_{n}\}$ $\leq$ lim sup $\{a_{n}\}$

Proof: let $\{a_n\}$ be a sequence such that $\{a_n\}$ is bounded, so by definition

lim inf $\{a_{n}\}$ = lim inf $\{a_{n}: n\geq k\}$ as n goes to infinity and,

lim sup $\{a_{n}\}$ = lim sup $\{a_{n}: n\geq k\}$ and, as n goes to infinity

thus i can conclude that lim inf $\{a_{n}\}$ $\leq$ lim sup $\{a_{n}\}$.

i feel like there is more to this problem than just using the definition to prove it...

Answer 1

Note that $$(\forall k\in \mathbb N) \inf_{n \ge k} a_n \le \sup_{n \ge k} a_n \implies \lim_{n \to\infty} \inf_{n \ge k} a_n \le \lim_{n \to\infty} \sup_{n \ge k} a_n.$$