An example of an unbounded increasing sequence that satisfies the assumptions of the convergence of monotone sequences...?
According to the convergence of monotone sequences if a sequences is monotonic and bounded then it converges to some $L$.
I think $S(n)=n+1$ since it's increasing and it's unbounded correct?
Your example $a_n = n$ satisfies that it is monotone but not bounded, and is therefore not necessarily convergent. For an example that is bounded but not monotone, just take the sequence $-1, 1, -1, 1, \ldots$. Any convergent sequence must be bounded. But not every convergent sequence must be monotone. Just take $a_n = (-1)^n/n$.