 Distance-increasing self-map theorem for sequentially compact metric spaces

Every continuous map $T:X\to X$ on a sequentially compact space with a metric satisfying $d(Tx,Ty)\ge d(x,y)$ is a surjective isometry.

A convergent subsequence of the paired orbit $(T^{k}x,T^{k}y)$ has two late terms arbitrarily close. Removing their common number of iterates by the distance inequality shows that $(x,y)$ returns arbitrarily close to itself after a common positive time $n$. Thus $d(Tx,Ty)\le d(T^{n}x,T^{n}y)\le d(x,y)+2ϵ$, and letting $ϵ\downarrow 0$ proves the isometry claim. For each $x$, choose return times $n_k\ge 1$ with $T^{n_k}x\to x$. A convergent subsequence of $T^{n_k-1}x$ has limit $z$, and continuity gives $Tz=x$. This proves surjectivity.