(i) Prove that if r is a rational number, then r - √√2 is irrational. (ii) For any two real numbers a and b, assume that the number of elements in the interval [a, b] is larger than the number of all rational numbers. Show that between any real numbers a and b, there must exist an irrational number. (Assume there is no irrationals between a and b and get a contradiction to the assumption.)
(i) Prove that if r is a rational number, then r - √√2 is irrational. (ii) For any two real numbers a and b, assume that the number of elements in the interval [a, b] is larger than the number of all rational numbers. Show that between any real numbers a and b, there must exist an irrational number. (Assume there is no irrationals between a and b and get a contradiction to the assumption.)
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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![(i) Prove that if r is a rational number, then r -. √2 is irrational.
(ii) For any two real numbers a and b, assume that the number of elements in the interval
[a, b] is larger than the number of all rational numbers. Show that between any real numbers
a and b, there must exist an irrational number. (Assume there is no irrationals between a
and b and get a contradiction to the assumption.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed14a3ea-da26-4be7-a143-8b845df95e91%2F80096fb6-b951-4f9c-91c2-3bc5c32014d2%2Fdbeylri_processed.png&w=3840&q=75)
Transcribed Image Text:(i) Prove that if r is a rational number, then r -. √2 is irrational.
(ii) For any two real numbers a and b, assume that the number of elements in the interval
[a, b] is larger than the number of all rational numbers. Show that between any real numbers
a and b, there must exist an irrational number. (Assume there is no irrationals between a
and b and get a contradiction to the assumption.)
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