Is it true that every open set contains a compact set?
Well, the empty set is open. And yes, even the empty set contains a compact set.
Showing that one open set contains a compact set does not prove what the OP wanted!
However, I can finish by showing the "easy" part: If A is not empty, then it contains, say, "a". The singleton set "a", since it is finite, is compact.
My point was, that every open set contains the empty open set, which contains the empty compact set. So I was not really doing *one* open set after all.
Separate names with a comma.