Let consider a normed vector space $V$. I want to prove that
If $f:V\to \mathbb R$ is a convex function and if for some $x_0 \in V$ the function is bounded on a neighborhood $W$ of $x_0$, then there exists a neighborhood $U$ of $x_0$ such that $f$ is Lipschitz on $U$.
When I say neighborhood of $x_0$ I mean an open set that contains $x_0$ WLOG the neighborhood clearly can be considered to be a ball with center $x_0$.
I don't know what can I do, because I can compute directly $f(x)-f(y)$ because the convexity only works with positive scalars. So I'm a little confused, please help me!