ILL 1987 Problem 76

Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \ (k \in \mathbb N)$ such that:(i) $a_k < b_k,$ (ii) $\cos a_kx + \cos b_kx \geq -\frac 1k$ for all $k \in \mathbb N$ and $x \in \mathbb R,$ prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit.

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