IMC 2014 · Problem 3

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