BG-MO 2008 Problem 3

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : $$\left|\sum_{i=1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}$$ for all $k\in\mathbb{N}$ . Prove that $b_1=b_2=\ldots =b_n=0$ .