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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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.
Transcribed Image Text: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 2.

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

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

Dedekind cut of irrational number 2 is that cut that consists of rational numbers that have squares less than 2.

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