KöMaL 2016 B4811

Prove that $\frac1{a_1} + \frac1{[a_1,a_2]} + \frac1{[a_1,a_2,a_3]} + \ldots + \frac1{[a_1,a_2,\ldots,a_n]} < 2,$ for all integers $0<a_1<a_2<\ldots<a_n$, where the symbol $[\dots]$ stands for the least common multiple.