I am struggling with a exercise from the book "Functional Analysis, Calculus of Variation and Optimal Control". The goal is to show the following claim:
A convex lower-semi-continuous (lsc) function $f: X \mapsto R_{\infty}$ on a Banach Space $X$ is locally Lipschitz on the interior of its domain.
There is also a solution hint:
Let $x_0$ be a point of the interior. It suffices to proof that the (closed) set: $C=\{ y \in X: f(x_0+y) \leq f(x_0)+1\}$ has nonempty interior.
a) First show that for every point $z \in X$ there exists $t>0$ such that $tz \in C$.
b) Second, invoke Baire's theorem to deduce that $int C \neq \emptyset$
My question refers to the last statement: Assume that I take the first statement $a)$ as established. In order to be able to apply Baire's theorem I need to find a countable collection of nowhere dense sets. And I am currently lacking the imagination on how to specify this countable collection (after all, I do not have any additional assumption on the separability of the space). Any hints/advices appreciated!
Note that by assumption, $C$ is closed. The statement of (a) can be strengthened to: for every $z\in X$ there is $t>0$ such that $sx\in C$ for all $s\in [0,t]$. Then we get $$ X = \bigcup_{n \in \mathbb N} (n C). $$ By Baire's theorem, one of the sets $nC$ has an interior point.