Exercise 11, § 2

Let $X$ be a compact metric space, with metric $d$. Show that if $f$ is a mapping of $X$ into $X$ such that, for each pair $x, y$ of points of $X$, we have $d(f(x), f(y)) = d(x, y)$, then $f$ is an isometry of $X$ onto itself. [Let $a, b$ be any two points of $X$, and put $f^n = f^{n-1} \circ f$, $a_n = f^n(a)$, $b_n = f^n(b)$; show that, for each $\varepsilon > 0$, there exists an index $k$ such that $d(a, a_k) < \varepsilon$ and $d(b, b_k) < \varepsilon$, by choosing suitable subsequences of $(a_n)$ and $(b_n)$; hence show that $d(a_1, b_1), \ldots d(a, b)$ and that $f(X)$ is dense in $X$.]