I understand that if a sequence is bounded by supremum and is strictly increasing it will converge. It is intuitive because the sequence is strictly increasing.
I do not really understand the weaker version where we say that any non-decreasing sequence bounded by supremum will converge. Suppose, I know that the sequence is always less than a value $\gamma$, so it is bounded. But if the sequence stops increasing at some point it will never get arbitrarily close to $\gamma$. Suppose the following situation:
$$S=\{1,2,3,4,5,6,...,10,10,10,10,10,10,10,10\}$$.
Suppose, I know that all elements of the sequence are less than 100. But the sequence gets stuck at some point because it is non-decreasing.