Let $X$ and $Y$ be two arbitrary subsets of $\mathbb{R}$. Show that $\overline{X\cap Y} \subseteq \overline{X} \cap \overline{Y}$
Proof
since $X\cap Y \subseteq X$ and $X\cap Y \subseteq Y$
$\implies \overline{X\cap Y} \subseteq \overline{X}$ and $\overline{X\cap Y} \subseteq \overline{Y}$
In other words, $\overline{X\cap Y}$ is present in both $\overline{X}$ and $\overline{Y}$. $$\implies \overline{X\cap Y} \subseteq \overline{X} \cap \overline{Y}$$ Is my proof correct?