Proving that $A\subset B \implies \hat A \subset \hat B$ where, $ \hat X $ implies closure of $X$. I want to prove this strictly using contradiction. So,I started out assuming $\exists x \in \hat A \text{ and } x \notin \hat B$
Since, $A \subset B, x\in (\hat A-A)$ (else it would be in B automatically).
Now, I need to somehow prove that any point in $\hat A - A$ either is a part of B or a part of $\hat B - B$.
Any leads? I don't want solution. This is homework.