What is $A\setminus U$ if $U$ is open and $A$ is closed?

Hint: rewrite $U\setminus A=U\cap A^c$. Now if $A$ is closed, what does that mean for the complement $A^c$?

$U \setminus A$ is, in words, all those elements of $U$ that aren't element of $A$. So if $A$ and $U$ were disjoint then this would be equal to $U$, but certainly not in general. If $A = X$ (where $X$ is the whole space) then no elements of $U$ wouldn't be elements of $A$ and the set would be empty. So it does depend on $A$.

Now $U \setminus A = U \cap (X \setminus A)$ (this is immediate from my description in words) and then we have written it as the finite intersection of an open set $U$ and the complement of a closed set $A$, so another open set. So the result must be open.

A set is closed iff its complement is open.

The intersection of two open sets is open.