How to prove by induction that $|\sin(nx)| \leq n|\sin x|$?
Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$ Now, here I got stuck. How to prove it?? Please help.
Solution 1:
The inductive step: Using the triangle inequality and the fact that $\sin$ and $\cos$ function are bounded by $1$ we get
$$|\sin((n+1)x)|=|\sin(nx)\cos x+\cos(nx)\sin(x)|\le|\sin(nx)|\cos(x)|+|\cos(nx)||\sin(x)|\\\le|\sin(nx)|+|\sin(x)|\le n|\sin(x)|+|\sin(x)|=(n+1)|\sin(x)|$$