Give an example of the following or prove that such a request is impossible: (b) A Dedekind cut corresponding to an irrational number. Write it without making explicit reference to the number. (e) An unbounded sequence containing a subsequence that is Cauchy.

Give an example of the following or prove that such a request is impossible: (b) A Dedekind cut corresponding to an irrational number. Write it without making explicit reference to the number. (e) An unbounded sequence containing a subsequence that is Cauchy.

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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Question

Analysis

Give an example of the following or prove that such a request is impossible:
(b) A Dedekind cut corresponding to an irrational number. Write it without making
explicit reference to the number.
(c) An unbounded sequence containing a subsequence that is Cauchy.

Expert Solution

Step 1

(a)

Consider the Dedekind cut of irrational number $\sqrt{2}$

A Dedekind cut is a subset of the rationals that satisfies:

It contains at least one number.
If x is in the set, any y with $y < x$ is also in the set.
The set contains no largest member.

Dedekind cut of irrational number $\sqrt{2}$ is that cut that consists of rational numbers that have squares less than 2.

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