Putnam 1967 A1

Let $f(x)= a_1 \sin x + a_2 \sin 2x+\cdots +a_{n} \sin nx$ , where $a_1 ,a_2 ,\ldots,a_n$ are real numbers and where $n$ is a positive integer. Given that $|f(x)| \leq | \sin x |$ for all real $x,$ prove that $$|a_1 +2a_2 +\cdots +na_{n}|\leq 1.$$