$|\frac{\sin(nx)}{n\sin(x)}|\le1\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$ Find all real number $n$ such that  $|\frac{\sin(nx)}{n\sin(x)}|\le1\forall x\in\mathbb{R}-\{\pi k: k\in\mathbb{Z}\}$.

Let $\phi_n(x) = |\frac{\sin(nx)}{n\sin(x)}|$. $\phi_0$ is undefined, so $n \neq 0$. Since $\phi_n = \phi_{-n}$, we can assume $n>0$.
Suppose $n>0$ is not an integer, and consider the sequence $x_k = \pi-\frac{1}{k}$. Then $x_k \to \pi$, but $n x_k \to n \pi \notin \pi \mathbb{Z}$. Consequently, $\lim_k \phi_n(x_k) = \infty$.
Now suppose $n>0$ is an integer, and
let $z = e^{ix}$. Then $\phi_n(x) = |\frac{z^{n}-z^{-n}}{n(z-z^{-1})}| = |\frac{z^n}{k z}\frac{1-z^{-2n}}{1-z^{-2}}| = |\frac{z^n}{k z} \sum_{k=0}^{n-1} (z^2)^k| $. Since $|z| = 1$, we have
$$\phi_n(x) \leq \frac{1}{k} \sum_{k=0}^{n-1} (1) = 1.$$
It follows that $\phi_n(x) \leq 1 $ for all appropriate $x$ iff $n$ is a non-zero integer.
