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9. We know that v2 is irrational. Use this fact and the previous exercise to prove that if r < s are rational numbers, then there is an irrational number x with r < x < s.

9. We know that v2 is irrational. Use this fact and the previous exercise to prove that if r < s are rational numbers, then there is an irrational number x with r < x < s.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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7. Prove that if x < y are two real numbers, then there is a rational number r with x <r < y. Hint: Use the result of Example 1.4.9. 8. Prove that if x is irrational and r is a non-zero rational number, then x+r and rx are also irrational. 9. We know that 2 is irrational. Use this fact and the previous exercise to prove that if r < s are rational numbers, then there is an irrational number x with r < x < s.
Transcribed Image Text:7. Prove that if x < y are two real numbers, then there is a rational number r with x <r < y. Hint: Use the result of Example 1.4.9. 8. Prove that if x is irrational and r is a non-zero rational number, then x+r and rx are also irrational. 9. We know that 2 is irrational. Use this fact and the previous exercise to prove that if r < s are rational numbers, then there is an irrational number x with r < x < s.
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