I'm working on the Exercise 1 from Chapter 2, in the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haïm Brezis. The exercise states as following:
Let $E$ be a Banach space and let $\varphi: E \rightarrow(-\infty,+\infty]$ be a convex l.s.c. function. Assume $x_0 \in \mathrm{Int}D(\varphi)$. Prove that there exists a constant $M$ and a closed ball $B_{f}(x_{0}, R)$, centered at $x_{0}$ with radius $R$, on which $\varphi \leq M$. Here $D(\varphi)$ is the set of points where $\varphi$ takes finite value.
My attempt: Let $x \in \mathrm{Int}D(\varphi)$. Let $\rho > 0$ such that $B_{f}(x, \rho) \subset \mathrm{Int}D(\varphi)$. For every $n \in \mathbb{N}$, define $F_{n} := \{ y \in E \mid \|y - x\| \leq \rho \text{ and } \varphi(y) \leq n \}$. Noting that $F_{n}$, $n \in \mathbb{N}$ are closed subset of $E$ and $B_{f}(x, \rho) \subset \cup_{n \in \mathbb{N}} F_{n}$, so $(F_{n})_{n \in \mathbb{N}}$ is a sequence of closed subsets of $E$ which union has non-empty interior. By Baire category theorem, at least one of $F_{n}$ has non-empty interior. So the set $S := \{ x \in \mathrm{Int}D(\varphi) \mid x \text { has a neighborhood on which } \varphi \text { is bounded above} \}$ is dense in $\mathrm{Int}D(\varphi)$. Moreover, we know $S$ is open and convex.