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:

Here's a partial solution,
which shows that
is it true for
$c = 1$.
Want to show that
$lcm[a_{k},a_{k+1}]>ck
$
infinitely often.
Since the $a_k$
are distinct,
$max(a_k)_{k=1}^n
\ge n
$.
Let $a_n$ be such that
$a_n = max(a_k)_{k=1}^n
$.
There are an infinite number of these,
since the $a_n$
are distinct positive integers.
For these $a_n$,
$a_n \ge n$.
Since
$lcm(a, b)
\ge max(a, b)
$,
$lcm(a_{n-1}, a_n)
\ge max(a_{n-1}, a_n)
\ge a_n
\ge n
$.
Right now,
I don't see how to
push $c > 1$.
If we could choose
an $a_{n-1}$ which does not
divide one of these extreme $a_n$
 infinitely often,
it looks like we count
get an unbounded $c$,
maybe.
