Erdős problem 25

Let $n_1 < n_2 < \dots$ be an arbitrary sequence of integers, each with an associated residue class $a_i \pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n < n_i$ or $n \not\equiv a_i \pmod{n_i}$. Must the logarithmic density of $A$ exist?