Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$ I am stuck on proofs with subsequences. I do not really have a strategy or starting point with subsequences. 
NOTE: subsequential limits are limits of subsequences
Prove: $a_n$ is bounded $\implies \liminf a_n \leq \limsup a_n$ 
Proof:
Let $a_n$ be a bounded sequence. That is, $\forall_n(a_n \leq A)$.
If $a_n$ converges then $\liminf a_n = \lim a_n = \limsup a_n$ and we are done.
Otherwise $a_n$ has a set of subsequential limits we need to show $\liminf a_n \leq \limsup a_n$:
This is where I am stuck...

Hint: Think about what the definitions mean. We have 
$$\limsup a_n = \lim_n \sup \{ a_k \textrm{ : } k \geq n\}$$
and $$\liminf a_n = \lim_n \inf \{ a_k \textrm{ : } k \geq n\}$$
What can you say about the individual terms $\sup \{a_k \textrm{ : } k \geq n\}$ and 
$\inf \{a_k \textrm{ : } k \geq n\}$ ?
