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This is a problem from ISI MStat 2018 PSA Problem 10 based on Dirichlet Function.

## Dirichlet Function – ISI MStat Year 2018 PSA Question 10

Let $x$ be a real number. Then $\lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right)$

• (A) does not exist for any x
• (B) exists for all x
• (C) exists if and only if x is irrational
• (D) exists if and only if x is rational

### Key Concepts

Limit

Sandwich Theorem

But try the problem first…

Source

ISI MStat 2018 PSA Problem 10

Introduction to Real Analysis by Bertle Sherbert

## Try with Hints

First hint

Check two cases separately one when x is rational and other is when x is irrational.

Second Hint

If $m!x$  is an integer, then $cos ^{2 n}(m ! \pi x) =1$

If x is rational $\frac{p}{q}$, then, eventually, for large enough m, m! will be divisible by q , so that $m!x$ will be an integer, and we have $\lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right) =1$

Final Step

If x is irrational, $m!x$  will never be an integer, and $|cos(m! {\pi } x)|<1$ , so that $\lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right) =0$  for all m>0 by Sandwich Theorem.