Explain what it means for a metric space $(M, d)$ to be (i) compact, (ii) sequentially compact. Prove that a compact metric space is sequentially compact, stating clearly any results that you use.

Let $(M, d)$ be a compact metric space and suppose $f: M \rightarrow M$ satisfies $d(f(x), f(y))=d(x, y)$ for all $x, y \in M$. Prove that $f$ is surjective, stating clearly any results that you use. [Hint: Consider the sequence $\left(f^{n}(x)\right)$ for $x \in M$.]

Give an example to show that the result does not hold if $M$ is not compact.