3. Prove that if (a, } and (b,} are Cauchy sequences of rational numbers satisfying lim (a, - b, = 0 and there exists e > 0 so that Jan >e and b>e for all n. then lim +- = 0.
3. Prove that if (a, } and (b,} are Cauchy sequences of rational numbers satisfying lim (a, - b, = 0 and there exists e > 0 so that Jan >e and b>e for all n. then lim +- = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![3. Prove that if \(\{a_n\}\) and \(\{b_n\}\) are Cauchy sequences of rational numbers satisfying
\[
\lim_{{n \to \infty}} |a_n - b_n| = 0
\]
and there exists \(c > 0\) so that
\[
|a_n| > c \quad \text{and} \quad |b_n| > c \quad \text{for all } n,
\]
then
\[
\lim_{{n \to \infty}} \left| \frac{1}{a_n} - \frac{1}{b_n} \right| = 0.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff906d8fa-9b6a-4046-ba2d-f959fe6282ae%2F7d41ea12-ce17-4dd4-86db-e04d69e3a10d%2Fhtx8v48_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Prove that if \(\{a_n\}\) and \(\{b_n\}\) are Cauchy sequences of rational numbers satisfying
\[
\lim_{{n \to \infty}} |a_n - b_n| = 0
\]
and there exists \(c > 0\) so that
\[
|a_n| > c \quad \text{and} \quad |b_n| > c \quad \text{for all } n,
\]
then
\[
\lim_{{n \to \infty}} \left| \frac{1}{a_n} - \frac{1}{b_n} \right| = 0.
\]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
