Q:

Is it possible to replace $\lim$ by $\sup$ here? 

Let $(a_m)$ be a monotonically increasing sequence.  Is it then possible that $\lim_{m\to\infty} a_m=\sup_m a_m$? 
I only know the fact that a bounded monotonically increasing sequence coverges to its supremum.  But in the question above, the sequence is not supposed to be bounded. 

A:

Two cases are possible: Either the sequence is bounded and in that case you know that the equation holds. 
Or the sequence is not bounded.  In that case, the sequence diverges and in fact tends to infinity, usually written as $\lim_{m\to\infty}a_m=\infty$. Also, the supremum of an unbounded set is usually denoted as $\infty$, so that with this convention (i.e., in the extended real) the given equality holds in all cases. 