Show that: $ \limsup n \{ n^2 \sqrt{2}\} = \infty $

Question

Show that:

$$ \overline{\lim} n \{ n^2 \sqrt{2}\} = \infty $$

Sorry for being so succinct: $n \to \infty$ and $0 < \{ n^2 \sqrt{2} \} < 1$ so I am not sure which effect should win out.

Hint use Tauber theorem. That's not much of a hint but that's all I have to go on.

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Checking the lim sup is not so difficult, but this does not constitute a proof:

Answer 1

For irrational positive $\alpha,$ the fractional part of $n^2 \alpha$ is equidistributed in the unit interval. This is Satz 13 in Weyl 1916.

In particular, it is larger than $1/2$ infinitely often.