Let $f\colon\mathbb R\to\mathbb R$ defined by
$$f(x)=a_1\sin x+a_2\sin2x+a_3\sin3x+\cdots+a_n\sin nx,$$ for some values $a_1,a_2,a_3,\cdots,a_n\in\mathbb R.$ Prove that $$ |f(x)|\le|\sin x|\quad \forall x\in \mathbb{R}$$ implies: $$|a_1+2a_2+3a_3+\cdots+na_n|\le1$$