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

Check the Answer


But try the problem first…

Answer: is (B)

Source
Suggested Reading

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.

ISI MStat 2018 PSA Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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