TITLE: Proving that a set is complete
QUESTION [7 upvotes]: I am working on a question and I am looking for some clarification. I can't seem to use what I know to complete the proof.

Let $\mathbf{x}_{0} \in \mathbb{R}^n$ and $R>0$. Prove that $U=\left \{ \mathbf{x} \in \mathbb{R}^n : \left \| \mathbf{x} - \mathbf{x_{0}} \right \| \leq R \right \}$ is complete.

What I know:

$U$ is complete if every Cauchy sequence of points in $U$ converges to a point in $U$.
If $\mathbf{x}_k$ is a Cauchy sequence then there is an integer $N$ such that 
$$||\mathbf{x}_k - \mathbf{x}_l|| < \epsilon$$
for all $k,l \geq N$
$U$ is the set of points $\mathbf{x} \in \mathbb{R}^n$ such that the distance between $\mathbf{x}$ and $\mathbf{x}_0$ is less than or equal to $R$.

A Cauchy sequence in $U$ will be made up of points $\mathbf{x}_{k,1},\mathbf{x}_{k,1},...,\mathbf{x}_{k,n}$ such that $|| \mathbf{x}_{k,1} - \mathbf{x}_{0}||$, $|| \mathbf{x}_{k,2} - \mathbf{x}_{0}||,..., || \mathbf{x}_{k,n} - \mathbf{x}_{0}||$ $\leq R$. It seems that if the sequence is made up of points in $U$ then it must converge to a point in $U$. Any help and clarification would be appreciated.

REPLY [2 votes]: I don't know how extensive your knowledge on compactness is, but assuming that you know some basic results, you may consider the following methodology. It is clear that $U$ is a bounded set. If you know how to prove that it is a closed set, too, then you can appeal to the Heine-Borel theorem to conclude that $U$ is a compact set. Then, using the fact that compact spaces are complete, you conclude the proof.