= Bounded convex functions are locally Lipschitz on an open ball
{title2=$|f(u)-f(v)|\leq L\|u-v\|$}
Let a real <convex function> satisfy $|f|\leq M$ on the open unit ball of a <normed space>. For $4r=1-\|x\|$, points $u,v\in B(x,r)$ satisfy $|f(u)-f(v)|\leq(M/r)\|u-v\|$. Extend the segment from $u$ through $v$ to length $2r$; its endpoint still lies in the domain, and <convexity> bounds the difference by $2M\|u-v\|/(2r)$. Reverse $u,v$ for the opposite inequality. Unlike finite-dimensional local continuity theorems, this argument applies in arbitrary dimension because boundedness on a full neighborhood is assumed.
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