Norm minimizer in a closed convex subset of a reflexive Banach space (${\min }_{x\in C}\Vert x\Vert$)

A nonempty norm-closed convex subset $C$ of a reflexive Banach space attains its distance to zero. For $d={\inf }_C\Vert x\Vert$, the sets $C\cap (d+1/n)B_X$ are nonempty weakly closed subsets of one weakly compact ball, and form a decreasing family. Compactness gives a point in their intersection, of norm $d$. Mazur theorem supplies weak closedness of $C$ and the balls; no sequential-compactness theorem is required.