Question: Let $(b\_n)$ be a monotone increasing sequence bounded above by $K$. Can you show that it converges to a limit?

Answer: 
$(bn)$ is bounded above $\implies \exists sup(b\_n)$ 

So $\forall \epsilon \gt 0 \implies \exists b\_N \in (b\_n) : sup(b\_n)-b\_N \lt \epsilon$

So $\forall n\ge N$ we have $sup(b\_n)-b\_n \le sup(b\_n)-b\_N \lt \epsilon$

Therefore, $\exists lim\_n {b\_n} = sup(b\_n)$.
If $(b\_n)$ doesn't converge to a limit, it must keep growing, and hence is not bounded above by $K$.