Proving that a compact space is regular I am studying a book and I am stagnating on a what should be a straightfoward proof:
Show that if $X$ is compact, $V\subset X$ is open and $x\in V$, then there exists an open set $U$ in $X$ with $x\in U\subset \bar U\subset V$
I don't know how to find the appropriate set $U$. I am guessing you need to do something like taking the intersection with $V$ of a finite subcover of X and then show that its closure is contained in $V$ ...
Could someone nudge me in the right direction?

Hint:
For any $y\in X\setminus V$ we have $x\neq y$, there exists $W_y$ a neighborhood of $y$ and $U_y$ a neighborhood  of $x$ s.t $U_y\cap W_y=\emptyset$. 
Now $W_y (y\in X\setminus V)$ is a cover of $X\setminus V$ which is compact (closed in Hausdorff compact space ), Let $W_{y_1},\ldots,W_{y_n}$ be a finite cover of $X\setminus V$.
Try with $U=\cap_{i=1}^nU_{y_i}$ this is a neighborhood of $x$.  
