8 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS or there exist 1 ≤ i j ≤ n so that α = [jα] − [iα] j − i . Thus, if the numbers are not distinct, the required inequality is trivially satisfied by α itself. Corollary 1.3.8. Given any real number α, there is a rational number p/q such that |α − p/q| 1/q2. Proof. This follows immediately from the theorem. Now comes the good news (or bad news depending on how you look at it). Theorem 1.3.9. If α is irrational, then there are infinitely many rational numbers p/q such that |α − p/q| 1/q2. Proof. Suppose there are only a finite number of rational numbers p1/q1, p2/q2, . . . , pk/qk satisfying the inequality. Then, there is a positive integer n such that |α − pi/qi| 1/(n + 1)qi for i = 1, 2,...,k. This contradicts The- orem 1.3.7, which asserts the existence of a rational number p/q satisfying q ≤ n and |α − p/q| 1/(n + 1)q 1/q2. So, there you have it, a real number α is rational if and only if there exists only a finite number of rational numbers p/q such that |α − p/q| ≤ 1/q2. Moreover, a real number α is irrational if and only if there exists an infinite number of rational numbers p/q such that |α − p/q| ≤ 1/q2. 1.4. Intervals At this stage we single out certain subsets of R which are called intervals. Definition 1.4.1. A subset of R is an interval if it falls into one of the following categories. (a) For a, b ∈ R with a b, the open interval (a, b) is defined by (a, b) = {x ∈ R | a x b}. (b) For a, b ∈ R with a ≤ b, the closed interval [a, b] is defined by [a, b] = {x ∈ R | a ≤ x ≤ b}. (c) For a, b ∈ R with a b, the half-open interval [a, b) is defined by [a, b) = {x ∈ R | a ≤ x b}. (d) For a, b ∈ R with a b, the half-open interval (a, b] is defined by (a, b] = {x ∈ R | a x ≤ b}. (e) For a ∈ R, the infinite open interval (a, ∞) is defined by (a, ∞) = {x ∈ R | a x}. (f) For b ∈ R, the infinite open interval (−∞,b) is defined by (−∞,b) = {x ∈ R | x b}. (g) For a ∈ R, the infinite closed interval [a, ∞) is defined by [a, ∞) = {x ∈ R | a ≤ x}.

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