Let and be such compacts. Take one point in
For every point of there are two open sets and such that and and .
so for a finite subset of .
Now is an open set disjoint from (it is disjoint from ) which contains
Repeat that operation for every You obtain a set of open sets such that and is disjoint from (more precisely from ).
So same properties (and even better) for for a finite subset of
what can you say of and ?