TIFR 2013 Problem 11 Solution - Cauchy sequence- series

TIFR 2013 Problem 11 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate program leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

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Problem:True/False

Consider the sequences \(x_n=\sum_{1}^{n} \frac{1}{j} \) and \(y_n=\sum_{1}^{n} \frac{1}{j^2} \). Then \(\left\{x_n\right\} \) is Cauchy but \(\left\{y_n \right\} \) is not.

Discussion: 

We are given sequence of partial sums of a very well known type of series.  \(\left\{x_n\right\}\) is a divergent sequence and \(\left\{y_n\right\}\) is convergent. Also, \(\mathbb{R}\) is complete. So every Cauchy sequence is convergent and any convergent sequence (as always happens in metric spaces) is Cauchy.

The true statement would be \(\left\{x_n\right\}\) is not Cauchy and \(\left\{y_n\right\}\) is Cauchy sequence.

Helpdesk

• What is this topic: Real Analysis
• What are some of the associated concept: Cauchy sequence, Convergent
• Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert