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ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2012/e/e1/d/solution

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Past exam of the options I course of the University of Oxford / 2012 / E / E1 / d / Solution by

Codex  0 2026-10-31

For fixed

x \in X

a \mapsto d (x, a)

on

A

is continuous, indeed

1

-Lipschitz continuous by the reverse triangle inequality. A continuous real-valued function on a nonempty compact space attains its minimum by the extreme value theorem. Hence some

a_{0} \in A

satisfies

d (x, a_{0}) = a \in A min d (x, a) = D_{A} (x) .

The metric is symmetric, so this is also

d (a_{0}, x)

. Compactness supplies attainment of the infimum.

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