Bounded convex functions are locally Lipschitz on an open ball ($|f(u)-f(v)|\le L\Vert u-v\Vert$)

Let a real convex function satisfy $|f|\le M$ on the open unit ball of a normed space. For $4r=1-\Vert x\Vert$, points $u,v\in B(x,r)$ satisfy $|f(u)-f(v)|\le (M/r)\Vert u-v\Vert$. Extend the segment from $u$ through $v$ to length $2r$; its endpoint still lies in the domain, and convexity bounds the difference by $2M\Vert u-v\Vert /(2r)$. 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.