Prove there $\operatorname{lcm}[a_{k},a_{k+1}]>ck$ 
Let $a_{1},a_{2},\cdots,a_{n}$ distinct postive integers .$0<c<\dfrac{3}{2}$

Show that:
there exsit infinitely postive integer $k$ such
$$\operatorname{lcm}[a_{k},a_{k+1}]>ck$$
From:2015 china TsT:

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The key is
$$\dfrac{1}{[a_{i},a_{i+1}]}\le\dfrac{1}{3}\left(\dfrac{1}{a_{i}}+\dfrac{1}{a_{i+1}}\right)$$
