How to prove an increasing sequence that converges is bounded above by its limit I am trying to prove that an increasing sequence that converges to $ L$ is bounded above by its limit.
By using  $a_n \le a_{n+1}$ and the definition of limit of a sequence, I can prove that for $\epsilon > 0$ , $ a_n \lt {L + \epsilon} $ for all $a_n$.
But is there a way to proceed to $ a_n \le L $ ? because I can't think of a case in which the former is true but the latter isn't.

If $a_n$ converges to $L$ and is increasing then let $\epsilon>0 \ ,\exists N \in \Bbb N \ s.t. \ \forall n>N \ \  \vert a_n-L\vert < \epsilon$
We can open up this up as you did to $a_n<L+\epsilon$
As indicated in the hint above, if $\exists n $ such that $L<a_n<L+\epsilon$ we get a contradiction because the limit is $L$ and so the sequence will have to "decrease" from $L<a_n<L+\epsilon$ down to $L$ 
