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.
Also Visit: College Mathematics Program of Cheenta
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.
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.
