Let formula_1 be a compact set in formula_2, and formula_3 a ball of radius 1 centered at formula_4. Then the set of all such balls centered at formula_5 is clearly an open cover of formula_1, since formula_7 contains all of formula_1. Since formula_1 is compact, take a finite subcover of this cover. This subcover is the finite union of balls of radius 1. Consider all pairs of centers of these (finitely many) balls (of radius 1) and let formula_10 be the maximum of the distances between them. Then if formula_11 and formula_12 are the centers (respectively) of unit balls containing arbitrary formula_13, the triangle inequality says: