This is the problem:
Let $X$ be a metric space with metric $d$ and $K\subset X$ compact and $F\subset X$ closed and $K\cap F=\varnothing$. Let $x\in F,y\in K$. Prove that there exists $\delta>0$ such that for every $x\in F$ and $y\in K$, $d(x,y)> \delta.$
My idea was to use sequences,once I know that in a compact set we have a subsequence converging to an element in $K$ and how $F$ is closed there's a sequence converging to an element in $F$.
But ,then I got stuck...I don't know what to do now.Any hint?Much appreciated!