State and prove the Bolzano-Weierstrass theorem in $\mathbb{R}^{n}$. [You may assume the Bolzano-Weierstrass theorem in $\mathbb{R}$.]

Let $X \subset \mathbb{R}^{n}$ be a subset and let $f: X \rightarrow X$ be a mapping such that $d(f(x), f(y))=d(x, y)$ for all $x, y \in X$, where $d$ is the Euclidean distance in $\mathbb{R}^{n}$. Prove that if $X$ is closed and bounded, then $f$ is a bijection. Is this result still true if we drop the boundedness assumption on $X$ ? Justify your answer.