16. Let A be the set of rational numbers in (0, 1). Since A is countable, it can be written as a sequence

????

i.e.,A = {rn : n = 1, 2, 3, . . .}

. Prove that for any ε > 0, A can be covered by a sequence of open balls whose total length is less than ε. That is, ∀ε > 0, there exists a sequence of open intervals (αn, βn) such that rn ∈ (αn, βn) and P

∞ n

=1(

βn −

αn)

<

ε. This important result explains why in a random selection of points from (0, 1) the probability of choosing a rational is zero.

Hint: Let αn = rn − ε/2n+2, βn = rn + ε/2n+2.