Using induction, prove $|\sin(nx)| \le n|\sin x|$ for $n\in \mathbb{N}$.
How I can show that with induction? Normally , When I make a induction I know for $n$, and I prove for $n+1$. But I can't write $n+1$ in this question. If I write $n+1$, formula change like this: $|\sin((n+1)x)| \le (n+1)|\sin x|$ for $n+1$, This new formula doesn't include $n$.
What should I do in such cases?
For induction step we can use that
$$|\sin ((n+1)x)|=|\sin(nx+x)|=|\sin(nx)\cos x+\sin x\cos(nx)|$$
$$\le |\sin(nx)\cos x|+|\sin x\cos(nx)|\le n|\sin x|+|\sin x|=(n+1)|\sin x|$$