The continuity of the best approximation projection onto a suitable subspace of a metric space is examined. Conditions are given for the existence of the infimum of the set of Lipschitz constants for a continuous projection on a compact subset of a metric space. It is also shown that the mapping taking a compact set to the corresponding infimum Lipschitz constant is uniformly continuous with respect to the Hausdorff metric. The paper is fully constructive: it uses intuitionistic logic throughout.
