Mazur-Ulam theorem

Every surjective isometry between real normed vector spaces is affine. In particular, if $u(0)=0$, then $u$ is real-linear.

For $v,w$ in a real normed space, start with the points at distance $\Vert v-w\Vert /2$ from both endpoints and repeatedly retain those within half the previous diameter of every retained point. Reflection about $(v+w)/2$ preserves every stage, the midpoint belongs to every stage, and the diameters decrease by a factor at least two. Their intersection is therefore exactly the midpoint.