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i have to find out the following limit $$\lim_{m\to\infty}\lim_{n\to\infty}[cos(n!πx)^{2m}]$$ for $x$ rational and irrational. for $x$ rational $x$ can be written as $\frac{p}{q}$ and as $n!$ will have $q$ as its factor the limit should be equal to 1.
the second part of irrational is giving me pr...
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Q: How to prove that $\lim(\underset{k\rightarrow\infty}{\lim}(\cos(|n!\pi x|)^{2k}))=\chi_\mathbb{Q}$
Possible Duplicate:
How is this called? Rationals and irrationals
Please help me prove, that
$$\underset{n\rightarrow\infty}{\lim}\left(\underset{k\rightarrow\infty}{\lim}(\cos(|n!\pi x|)^{2k})\right)=\begin{cases}
1 & \iff x\in\mathbb{Q}\\
0 & \iff x\notin\mathbb{Q}
\end{cases}$$
See...
