CIIM 2014 Problem 4

Let $\{a_i\}$ be a strictly increasing sequence of positive integers. Define the sequence $\{s_k\}$ as $$s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},$$ where $[a_i,a_{i+1}]$ is the least commun multiple of $a_i$ and $a_{i+1}$ . Show that the sequence $\{s_k\}$ is convergent.