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TIFR 2013 Problem 34 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 programe 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?

Let $S$ be the set of all sequences $\{a_1,a_2,…,a_n,…\}$ where each entry $a_i$ is either $0$ or $1$. Then $S$ is countable.

## Hint:

What if instead of $0$ and $1$ the values were allowed to be any digit from $0$ to 9? What would that correspond to?

## Discussion:

Given a sequence  $\{a_1,a_2,…,a_n,…\}$ , we can associate it to the binary number $0.a_1a_2…a_n…$. This association (or function if you like) is one-one, and onto the set of all real numbers having binary expansion in the form of $0.something$, which is same as the set $(0,1)$, which is uncountable.

## Remark:

One could use Cantor’s diagonalization argument as well to argue in this problem. If possible let $x_1,x_2,…$ is an enumeration of the given set (of sequences) then consider the sequence $\{b_1,b_2,…\}$ in $S$ defined by $b_i \ne x_i$. We get a contradiction because this is a sequence which is not in the enumeration but is a member of $S$.

## Helpdesk

• What is this topic: Real Analysis
• What are some of the associated concept: Cantor’s diagonalization argument, Sequence, Sequential criterion
• Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert