(Sheet 4) Pudding - Limit of limits

Let \( x \) be a real number.  Let \( a_{m,n} = \cos(\frac{m}{n} \pi x) \).

What can you say about \( \lim_{m\to \infty} (\lim_{n \to \infty} a_{m,n}) \)?

Now consider \( \lim_{n\to \infty} (\lim_{m\to \infty}a_{m,n}) \). Does this iterated limit always exist?  Exist for some values of \( x \) but not others? Never exist?

What conclusions do you draw?