# How to prove <munder> <mo movablelimits="true" form="prefix">lim&#x2006;sup <mrow clas

How to prove $\underset{n\to \mathrm{\infty }}{lim sup}|\mathrm{sin}\left(n\right)|=1$?
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poquetahr
See this article, a special case is that there is two increasing sequences of odd positive integers $\left({p}_{n}\right),\left({q}_{n}\right)$ such that
$|\frac{\pi }{2}-\frac{{p}_{n}}{{q}_{n}}|\le \frac{1}{{q}_{n}^{2}},\phantom{\rule{1em}{0ex}}n>1.$
Note that $\mathrm{sin}|x|=|\mathrm{sin}x|$ for $x\in \left[0,\pi \right]$, then
$\mathrm{sin}|\frac{{q}_{n}\pi }{2}-{p}_{n}|=|\mathrm{cos}{p}_{n}|<\frac{1}{{q}_{n}}\to 0.$
therefore $|\mathrm{sin}{p}_{n}|\to 1$