The indicator function of a closed set $C \in \mathbb{R}^n$ is defined as bellow: $$ \delta_{C}(x)= \begin{cases} 0 \quad \quad\,\, x \in C\\ + \infty \quad x \notin C\\ \end{cases}. $$
Question
Is the indicator function of a closed set prox-bounded?
Prox-boundedness[Exercise 1.24 of Variational analysis]
A proper, lower semi-continuous function $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ is prox-bounded if for some $r \in \mathbb{R}$, $f(x)+\frac{1}{2}r \|x\|^2$ is bounded from below.
My try
According to this solution the indicator function of a closed set is lower semi-continuous but I am not sure whether it is proper. If we let $f(x):=\delta_{C}(x)$ we need to show $\delta_{C}(x)+\frac{1}{2}r \|x\|^2$ is lower bounded for some fixed $r$ which looks like it is since the lower bound of both is zero.