Proving a sequence is a Cauchy sequence from other theorems

Question

I would like to know if this is an acceptable proof. I have the following statement

Show directly that a bounded, monotone increasing sequence is a Cauchy sequence.

Let $\{x_n\}$ be a sequence that is bounded and monotone increasing. Then, by the Monotone Convergence Theorem, $\{x_n\}$ converges. Furthermore, by the Cauchy Criterion, we know that every convergent sequence is a Cauchy sequence and so we are done. QED

Does this work?

Answer 1

Let $L=\sup_{n}x_{n}$, for $\epsilon>0$, find some $N$ such that $L-\epsilon<x_{N}\leq L$. As $\{x_{n}\}$ is increasing, $L-\epsilon<x_{N}\leq x_{n}\leq L$ for all $n\geq N$. In particular, $|x_{n+p}-x_{n}|=x_{n+p}-x_{n}=x_{n+p}-L+L-x_{n}\leq L-x_{n}<\epsilon$ for all $n\geq N$ and $p=1,2,...$

Answer 2

Start here. Put $s = \sup_n x_n$; show that $x_n \to s$.