# Prove: Monotonic And Bounded Sequence- Converges

Let $a_n$ be a monotonic and bounded sequence, WLOG let assume it is monotonic increasing.
$a_n$ is bounded therefore there is a Supremum, $Sup(a_n)=a$, therefore $a_n<a+\epsilon$.
On the other hand due to $Sup(a_n)=a$, there is $N$ such that $a-\epsilon<a_N\leq a_n$ and together $a-\epsilon<a_N\leq a_n<a+\epsilon$ therefore $a-\epsilon< a_n<a+\epsilon$ and $lim_{n\to \infty}a_n=a$

Is the proof valid? does it apply to strictly monotonic sequence too?