real analysis - How to prove every Cauchy Sequence in $mathbb{R}^n$ converges

This is an analysis exercise that I have been struggling with for some time now. I am not familiar with metric spaces.

In $\mathbb{R}$, the book that I am using proves this fact by showing that every Cauchy sequence in $\mathbb{R}$ is bounded. Next, they use Bolzano-Weierstrass to choose a convergent subsequence of that Cauchy sequence.

However, the book does not specify an analog to boundedness in $\mathbb{R}^{n}$. Also, the book proved Bolzano-Weierstrass for $\mathbb{R}$, not $\mathbb{R}^{n}$. I was originally planning to outline the $\mathbb{R}$ approach by proving boundedness and choosing a convergent subsequence, but this is not currently possible because of what I said.

I was wondering if there is a good way to do this problem.

Thanks