Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be such that $\left\| f(x)-f(y)\right\| =\left\| x-y\right\|$. Is $f$ necessarily surjective?

If this is so, you can prove (Mazur-Ulam Theorem) that $f$ is affine, and hence you could classify all isometries of $\mathbb{R}^n$. However, at the moment, I can't think of any good ideas to prove that $f$ is surjective. For that matter, is it even the case that $f$ must be surjective?

Any ideas would be most welcomed.

Thanks much!