Isn't this the Monotone convergence theorem?
"If a sequence is monotone and bouned, then is converges"
here is an outline:
Suppose the sequence is increasing and bounded above.
Consider the set of all points in the sequece .
By the axiom of completeness, exists (since the set is a bounded subset of .
Assume , where .
Then, for , . Thus, there exists a point such that . Furthermore, because our sequence is increasing, it follows that, for every , . Thus, .
This implies