Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Problem 11.2.19. Since the convergence of Cauchy sequences can be taken as the completeness axiom for the real number system, it does not hold for the rational number system. Give an example of a Cauchy sequence of rational numbers which does not converge to a rational number.

Problem 11.2.19. Since the convergence of Cauchy sequences can be taken as the completeness axiom for the real number system, it does not hold for the rational number system. Give an example of a Cauchy sequence of rational numbers which does not converge to a rational number.

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
See similar textbooks
icon
Related questions
Question

Real math analysis, I need help with problem 11.2.19.

thank you so much.

Problem 11.2.19. Since the convergence of Cauchy sequences can be taken as the completeness axiom for the real number system, it does not hold for the rational number system. Give an example of a Cauchy sequence of rational numbers which does not converge to a rational number.
Transcribed Image Text:Problem 11.2.19. Since the convergence of Cauchy sequences can be taken as the completeness axiom for the real number system, it does not hold for the rational number system. Give an example of a Cauchy sequence of rational numbers which does not converge to a rational number.
Theorem 11.2.15. Cauchy sequences converge Suppose (sn) is a Cauchy sequence of real numbers. There exists a real number s such that limn→∞ Sn = s. Sketch of Proof. We know that (sn) is bounded, so by the Bolzano-Weierstrass Theorem, it has a convergent subsequence (Snk) converging to some real number s. We have sn – s = |Sn – Snp + Snk s| < |Sn – Sni|+|Snk 8|. If we choose n and ng large enough, we should be able to make each term arbitrarily small.
Transcribed Image Text:Theorem 11.2.15. Cauchy sequences converge Suppose (sn) is a Cauchy sequence of real numbers. There exists a real number s such that limn→∞ Sn = s. Sketch of Proof. We know that (sn) is bounded, so by the Bolzano-Weierstrass Theorem, it has a convergent subsequence (Snk) converging to some real number s. We have sn – s = |Sn – Snp + Snk s| < |Sn – Sni|+|Snk 8|. If we choose n and ng large enough, we should be able to make each term arbitrarily small.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS