3. We show that between any two distinct real numbers, there must exist a rational number. a 1 a. Suppose a and b are re

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3 We Show That Between Any Two Distinct Real Numbers There Must Exist A Rational Number A 1 A Suppose A And B Are Re 1 (99.72 KiB) Viewed 15 times

3. We show that between any two distinct real numbers, there must exist a rational number. a 1 a. Suppose a and b are real numbers with 1 < b - a. Prove that there must be an integer k such that a < k < b. (One approach: Let m be the smallest integer such that b < m. What can you say about m 1?) b. Use part (a) to show that if a' and b' are real numbers and n is a positive integer k' with = < b' – a', then there exists an integer k' such that a' < Show that given two real numbers a' < b', that there is always a positive integer n such that = <b' – a'. (One approach: Consider the quantity d. Conclude that for any real numbers a' < b' that there must be a rational number on the open interval (a', b'). <b'. n n C. 1 1 n b'-a'