IMC 2014 Problem 8

Let $f(x)=\frac{\sin x}{x}$ , for $x>0$ , and let $n$ be a positive integer. Prove that $|f^{(n)}(x)|<\frac{1}{n+1}$ , where $f^{(n)}$ denotes the $n^{\mathrm{th}}$ derivative of $f$ .

(Proposed by Alexander Bolbot, State University, Novosibirsk)