Let $X$ be a topological space and $Y$ be a metric space. Prove that the space of continuous functions $C(X, Y )$ with the topology of compact convergence is Hausdorff. 

The compact convergence topology is generated by the collection of sets $B\_C(f,ϵ)=\{g∈Y^X: \text{sup }\_{x∈C} d(f(x),g(x))<ϵ\}$, with compact $C⊂X$.

I know to show a topology is Hausdorff we need to show that for any two distinct elements in the topology we can find neighborhoods of those elements that do not intersect, but I am not sure how to do this for the compact convergence topology. The functions and supremum are throwing me off.