Show that the closure $\overline E$ of a subset $E \subset \mathbb R^n$ is the smallest closed set containing $E$.

I am new to topology so I was wondering how to establish a solid proof of this statement that seems basic.

Here is my attempt:
First we can show that $\overline E$ is a closed set containing $E$ :

* $\overline E = E \cup \partial E $ so it surely contains $E$.
* $\overline E^C = E^C \cap (\partial E)^C$. I need to show that $\forall \vec{x} \in \overline E^C, \exists \delta\_x > 0 \text{ such that } B(\vec{x},\delta\_x) \subset \overline E^C$ but I struggle to show it properly.
* Then I need to show that $\overline E$ is the smallest set with the previous conditions. I wanted to do it by contradiction by supposing that $\exists F \subset \overline E$ (so smaller than $\overline E$) and then find a contradiction but it seems too strange to prove.