SCHWEITZER 1978 Problem 3

Let $1<a_1<a_2< \ldots <a_n<x$ be positive integers such that $\sum_{i=1}^n 1/a_i \leq 1$ . Let $y$ denote the number of positive integers smaller that $x$ not divisible by any of the $a_i$ . Prove that $$y > \frac{cx}{\log x}$$ with a suitable positive constant $c$ (independent of $x$ and the numbers $a_i$ ).

I. Z. Ruzsa