Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / A / 1 / a

Assume $A$ is nonempty, as is necessary for an approximating point to exist. The reverse triangle inequality givesThus the distance from the fixed point $u$ is a continuous, indeed Lipschitz continuous, real-valued function on $A$. A continuous function on a nonempty compact set attains its minimum, so some $p\in A$ satisfiesIf empty compact sets are allowed, the printed statement needs this nonemptiness qualification.