nt.number theory – Is the factorization of $a_m-a_n$ affected by the fact that $Sigma frac{1}{a_k}

I would like to ask the following.

Let $(a_n)$ be a sequence of natural numbers such that
$sum_{k=1}^{infty}frac{1}{a_k}$ converges. Is it true that for
infinitely many $m$, there is a $n<m$ such that $a_m-a_n$ has a prime
divisor greater than $m$?

In other words, is it true that if for every $m, n$, the difference $a_m-a_n$ has all it’s prime factors less than or equal to $m$, then $sum_{k=1}^{infty}frac{1}{a_k}=+infty$?