The Real Number System in an Algebraic Setting by J. B. Roberts

Author:J. B. Roberts [Roberts, J. B.]
Language: eng
Format: epub
Publisher: INscribe Digital

Let a, b be in A, B, respectively, with b2 > r. Then a2 < r < b2. Let m′ be in Z, and let r1, r2, · · ·, rn be a finite chain from a to b such that

(Note that n depends upon m′.) Now there is a smallest i for which . Let j be this smallest i. Then

Now since rj − rj−1 < 1/m′, we have

and therefore

By exercise 10, p. 53, if we choose m′ sufficiently large, then . Since , this completes the proof.

COROLLARY 1. Between any two numbers in R there is a square of a number in R. That is, the set of squares of elements of R is dense in R with respect to <.

COROLLARY 2. If r has no square root in R, then the class A us defined in the proof of the lemma has no greatest number, and the class B has no least number.

These corollaries are to be proved in exercises 5 and 6, below.

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