Each has an open neighbourhood with for all n>1. Let , and use Urysohn or Tietze to get an open set with

Now use induction. Suppose that open sets with disjoint closures have been chosen so that and for and for all Show that each has an open neighbourhood disjoint from for all r>n, and also disjoint from As for the case n=1, define open sets

and check that has the right properties to continue the induction.