convergence divergence - 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.

Answer

HINT

You can easily show that if for some n $a_n>L$ then by definition of limit $a_n$ must decrease which is impossible.

You only need to formalize this idea by setting “assume exists n such that ...then by definition of limit...contradiction”.

Notably

• suppose $\exists n_1$ such that $a_{n_1}>L$ with $d=a_{n_1}-L>0$


• set $\epsilon=d$ by definition of limit must exists $n_2>n_1$ such that $|a_{n_2} -L|<\epsilon \implies a_{n_2}