A trigonometric inequality

Statement

Induction is often used to prove inequalities. As an example, we prove that $\left|\sin nx\right| \leq n\left|\sin x\right|$ for any real number $x$ and natural number $n$.

At first glance, it may appear that a more general version, $\left|\sin nx\right| \leq n\left|\sin x\right|$ for any real numbers $n,x$, could be proven without induction; but the case $n = \frac{1}{2},\, x=\pi$ shows it may be false for non-integer values of $n$. This suggests we examine the statement specifically for natural values of $n$, and induction is the readiest tool.