calculus - 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_nOn the other hand due to $Sup(a_n)=a$, there is $N$ such that $a-\epsilontherefore $a-\epsilon< a_n

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