Let A be the set of rational numbers in (0, 1). Since A is

Chapter , Problem 14

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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 > 0, A can be covered by a sequence of open balls whose total length is less than . That is, > 0, there exists a sequence of open intervals (n, n) such that rn (n, n) and / n=1(nn) < . This important result explains why in a random selection of points from (0, 1) the probability of choosing a rational is zero. Hint: Let n = rn /2n+2, n = rn + /2n+2.

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