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杉浦光夫16page問題2)

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$$ f(x) = \lim_{n \to \infty} \left [ \lim_{m \to \infty} \left \{ {\rm cos} \left ( n!\pi x \right ) \right \} ^{2m}\right ] $$

i) $x$が無理数の時、かつ$n$が有限の時
$n!$は有限の自然数で、$n!x$は無理数。よって、$ \left \{ {\rm cos} \left ( n!\pi x \right ) \right \} ^2 < 1$。
よって、$\lim_{m \to \infty} \left \{ {\rm cos} \left ( n!\pi x \right ) \right \} ^{2m} = 0$。

ii) $x$が無理数の時、かつ$n$が無限($n \to \infty $)の時
$n!$が有限の自然数ならば、$\left ( n +1 \right )!$ も有限の自然数。よって、$\left ( n+1 \right )!x$は無理数。よって、$ \left [ {\rm cos} \left \{ \left ( n + 1 \right )!\pi x \right \} \right ] ^2 < 1$。よって、$\lim_{m \to \infty} \left [ {\rm cos} \left \{ \left ( n+1 \right ) !\pi x \right \} \right ] ^{2m} = 0$。よって、全ての自然数$n$に対して、$\lim_{m \to \infty} \left \{ {\rm cos} \left ( n!\pi x \right ) \right \} ^{2m} = 0$。よって、$f \left ( x \right ) = 0$

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