Proof about limit of a monotone increasing sequence bounded above Let $(b_n)$ be a monotone increasing sequence bounded above by $K$. Can you show that it converges to a limit?

$(bn)$ 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, $\exists lim_n {b_n} = sup(b_n)$.
