I came across this problem in an old set of class notes:
Let $A \subset \mathbb{R}$ be a nonempty, bounded set. Let $\alpha = \sup{A}$, and let $(a_n)$ be a convergent sequence in $A$, with $a = \lim{a_n}$. Give an example where $(a_n)$ is strictly increasing, yet $a \ne \alpha$.
It seems to me that a counter-example does not exist. Otherwise, wouldn't this violate Monotone Convergence Theorem?
As the question stands it can't be solved, but not because of the issues you suspect. If $A$ is finite, then there simply are no strictly increasing sequences in it at all. Even if $A$ is infinite, it does not guarantee the existence of such a sequence as requested. For instance, in the set $A=\{-1/n\mid n\ge 1\}\cup \{0\}$ all strictly increasing sequences converge to the least upper bound, $0$. It would appear that by "give an example ...." it is meant that you are free to choose $A$ as well (the question is poorly worded). Then $A=[-1,1]$ and the sequence $a_n=-1/n$ works.