Cute convergence problem. Proving convergence of sequence regarding reciprocals of least common multiple converges. This is the first problem of the second day of the $2014$ CIIM. 
Let $\{a_n\}$ be a strictly increasing sequence of positive integers. 
Prove the sequence $\sum\limits_{i=0}^n\frac{1}{[a_i,a_{i+1}]}$ converges, where $[m,n]$ is the least common multiple of $m$ and $n$.
Hint:

 Try to bound the summands with the summands of a telescoping sum.


Note that $[a_i, a_{i+1}](a_i, a_{i+1}) = a_i a_{i+1}$ and consequently,
$$\sum_{i=0}^{n} \frac{1}{[a_i, a_{i+1}]} = \sum_{i=0}^{n} \frac{(a_i,a_{i+1})}{a_ia_{i+1}}\le \sum_{i=0}^n \frac{a_{i+1}-a_i}{a_i a_{i+1}} = \sum_{i=0}^n \frac{1}{a_i}-\frac{1}{a_{i+1}} = \frac{1}{a_0} - \frac{1}{a_n} < \frac{1}{a_0},$$
so the infinite sum converges.
