I need to prove using definition, I tried to prove that the limit isn't $0.5$ by adding and subtracting and finding some $n$ using floor function but it doesn't work. Any help? Thank you
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1The limit as $n \to \infty$? – Henry Nov 09 '20 at 10:34
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Please read this tutorial about how to typeset mathematics on this site. – N. F. Taussig Nov 09 '20 at 10:36
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1You could choose, for example, $n=2\pi k$ for integer $k$. Then $\sin(n)=\sin 2\pi k$ definitely doesn't approach $\frac{1}{2}$ ... – Matti P. Nov 09 '20 at 10:42
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limit as n→∞ and n is natural – ASyoliSA Nov 09 '20 at 10:43
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2The limit doesn't exist. Technically, $(\sin{n})_{n\in\mathbb{N}}$ is dense in $[-1,1]$. – rtybase Nov 09 '20 at 10:54
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Alternatively, you can find a subsequence converging to $0$. – rtybase Nov 09 '20 at 10:58
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That what I need to prove – ASyoliSA Nov 09 '20 at 10:58