Let $A$ be a metric subspace of $\mathbb{R}^n$ where we have the euclidean metric.
How do I prove that $\overline{A}$ with the inclusion map $f:A\hookrightarrow\overline{A}$ is a completion of $A$?
What I have done:
To be completion we must prove all of the following three characteristics:
1 $\overline{A}$ is complete
2 $f$ is an isometry
3 $f(A)$ is dense in $\overline{A}$
As stated in the comments by @Augustin and @Crostul $\overline{A}$ is the smallest closed subset of $\mathbb{R}^n$ that includes $A$, hence $\overline{A}$ is complete as well.
What I'm not sure about is: isn't $f$ the identity in this case, which would mean that it is an isometry?
And how do I prove 3? I know $f(A)=A$, so the closure of $f(A)$ in $\overline{A}$ is simply $\overline{A}$. Is this correct?