Supremum of $\cos(n)/n$ for $n\geq 1$

Question

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?