1. Assume for contradiction that is an extreme point that does not have this property, and define and and define two points:
Clearly and . We now show that for a small enough , and are feasible:
Consider a constraint then for small enough epsilon this constraint is still feasible. Now consider a constraint if the constraint is not tight then for small enough the constraint is still feasible, if the constraint is tight then either (w.l.o.g) and the constraint is still feasible or and the constraint is still feasible.
2. Consider an extreme solution, , to the LP and a 4-coloring of the vertices. Define and and let the partition of according to the 4-coloring. It is easy to see that plus any 3 sets out of form a feasible vertex cover. If we take the three sets out of with the minimum cost then the total cost of our solution is at most the value of the LP and thus a -approximation algorithm.