Proving $ \sum_{i=1}^{n-1} \frac{1}{lcm(a_{i} , a_{i+1})} \lt 1 $ for a set of increasing positive integers.

Question

assume $a_{1}, a_{2},...,a_{n}$ are numbers $\in \mathbb {N}$ such that $ a_{1} \lt a_{2} \lt... \lt a_{n} $; prove: $$ \sum_{i=1}^{n-1} \frac{1}{lcm(a_{i} , a_{i+1})} \lt 1 $$

Answer 1

$$\sum_{i=1}^{n-1}\frac{1}{\text{lcm}(a_{i},a_{i+1})} = \sum_{i=1}^{n-1}\frac{(a_{i},a_{i+1})}{a_{i}a_{i+1}} \leq \sum_{i=1}^{n-1}\frac{a_{i+1}-a_{i}}{a_{i}a_{i+1}} = \sum_{i=1}^{n-1}\frac{1}{a_{i}}-\frac{1}{a_{i+1}} = \frac{1}{a_{1}}-\frac{1}{a_{n}}<1$$