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ISI MStat 2018 PSA Problem 10 | Dirichlet Function

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

ISI MStat 2018 PSA Problem 10

Introduction to Real Analysis by Bertle Sherbert

Try with Hints

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

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 $$\lim {m \rightarrow \infty}\left(\lim {n \rightarrow \infty} \cos ^{2 n}(m ! \pi x)\right) =1$$

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.

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