Prove that a nest of sets has an empty intersection Let $f$ be a real convex function and $S$ an arbitrary closed bounded subset of the relative interior of the effective domain of $f$. Let $B$ be a closed Euclidean unit ball. The nest of sets
$$(S + \varepsilon B) \cap (\mathbb{R}^n \backslash \text{int}(\text{dom} f)), \quad \varepsilon > 0$$ has an empty intersection. Why? Is it because $\text{cl}S=\bigcap \{S + \varepsilon B \big | \varepsilon > 0\}$ and $S \subset \text{int}(\text{dom} f))$? 

Notice that
$$\bigcap_{\varepsilon>0} \left((S+\varepsilon B) \cap (\mathbb R^n\setminus\operatorname{int}(\operatorname{dom} f))\right)
= \left(\bigcap_{\varepsilon>0} (S+\varepsilon B)\right) \cap (\mathbb R^n\setminus\operatorname{int}(\operatorname{dom} f)).$$
As you said, the intersection appearing on the RHS is simply the closure, $\overline S=\bigcap_{\varepsilon>0} (S+\varepsilon B)$. And since $S$ is closed, we have $S=\overline S$.
So we want to show in fact that
$$S\cap (\mathbb R^n\setminus\operatorname{int}(\operatorname{dom} f))$$
which is equivalent to
$$S \subseteq \operatorname{int}(\operatorname{dom} f).$$
