= Norm minimizer in a closed convex subset of a reflexive Banach space
{title2=$\min_{x\in C}\|x\|$}
A nonempty norm-closed convex subset $C$ of a <reflexive Banach space> attains its distance to zero. For $d=\inf_C\|x\|$, 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.
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