How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

TIFR 2013 Problem 16 Solution -Bounded or Not?

TIFR 2013 Problem 16 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 programme 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


Suppose \(\left \{a_i\right\}\) is a sequence in \(\mathbb{R}\) such that \(\sum|a_i||x_i|<\infty\) whenever \(\sum|x_i|<\infty\). Then \(\left \{a_i\right\}\) is a bounded sequence.


For any \(r\in(0,1)\), \(\sum r^n <\infty \).

Also, if the radius of convergence of a power series is R, then R is  given by \(limsup|a_n|^{1/n}=\frac{1}{R} \)


Of course, \(\sum|a_n||r|^{n}<\infty\) for any \(r\in(-1,1)\).

Recall that, \(\sum|a_n|x^{n}<\infty\) for \(|x|<m\) means that radius of convergence of the power series is atleast m.

If the radius of convergence of  \(\sum|a_n|x^{n}\) is R then \(R\ge1\).

i.e, $$  limsup|a_n|^{1/n} = \frac{1}{R} \le 1  $$

Hence, there exists \(N\in \mathbb{N}\) such that \(sup\{|a_n|^{1/n} : n \ge k\} \le 1 \) for all \(k \ge N\) (This is from the definition of limsup of a sequence).

Hence,  \(sup\{|a_n|^{1/n} : n \ge N\} \le 1 \). Therefore, for each \(n \ge N\) we have \(|a_n|^{1/n} \le 1 \) for all \(n \ge N\). (Because sup is supremum which is least upper bound).

A real number which is in between 0 and 1 when raised to any power stays in between 0 and 1.

This allows us to state that \(|a_n| \le 1\) for all \(n \ge N\).

There are only finitely many terms left in the sequence which may not bounded by 1. But taking the maximum of their absolute value and 1 together we get a bound for the whole sequence.

For any \(n\in \mathbb{N}\),

$$ |a_n| \le max\{1,|a_1|,|a_2|,...,|a_{N-1}| \} $$


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

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.