I am looking to prove that if U is compact and V is closed, with UnV (intersection) is empty, then d(U,V) gt; 0 (the distance between the sets). We...

I am looking to prove that if U is compact and V is closed, with UnV (intersection) is empty, then d(U,V) > 0 (the distance between the sets).We define the distance between the sets as d(U,V) := {|x-y|:x in U, y in V}.Looking for help from someone comfortable in analysis/topology.