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How to find the supremum of $\{\cos(n)/n:n\in\mathbb{N}\}$? When I draw a graph of $x\mapsto \cos(x)/x$, it would be $\cos(1)$ that is the supremum. But, I can't prove it rigoriously, as it decreases and increases in some intervals due to periodicity. Is there any way to define a sequence in terms of $\cos(n)/n$ in order to use the monotone convergence theorem?

Mr.MathDoctor
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    do some case work. For smaller $n$, you can find the supremum by just looking at the values and after some point the denominator takes over and makes it too small. – dezdichado Sep 18 '22 at 20:44
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    Note that $\cos 1 \ge 1-{1^2 \over 2!} = {1 \over 2}$, $\cos n < 0$ for $n=2,3,4$ and so for $n>4$ we have ${ \cos n \over n} \le {1 \over 5} < {1 \over 2}$. – copper.hat Sep 18 '22 at 20:51

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