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Here “isometry” is used in the standard Mazur-Ulam theorem sense of a surjective distance-preserving map. Surjectivity is needed; a merely distance-preserving embedding need not preserve midpoints, as shown by a nonsurjective isometry need not preserve midpoints.

Distance preservation and surjectivity giveAssume inductively thatThen the two sets have the same diameter, and the universal distance condition defining the next set transfers through the bijection $u$. Hencefor every $n$.

Because $u$ is injective, it also preserves the intersection of this nested family. Part (a) therefore givesThus