Exercise: Suppose that we have an arbitrary metric space $(M,d)$ where $d$ is the Euclidian metric. If $(A,d)$ is complete, show that $A$ is closed in $M$.
I know that:
- $(A,d)$ is complete if every Cauchy sequence in $A$ converges to a point in $A$.
- If a sequence $(x_n)\subset A$ converges to some point $x\in M$, implies $x\in A$, then $A$ is closed.
So what I want to do is show that if every Cauchy sequence in $A$ converges to a point in $A$, then every sequence in $A$ converges to a point in $A$. I'm not sure how though. A sequence is Cauchy if given $\epsilon >0, \exists N>0$ such that $d(x_n, x_m) < \epsilon$ when $m,n\geq N$, and I'm not sure how to proceed from here.
Question: Is my approach going somewhere? How should I proceed to solve this exercise?