Erdős Problem #281

Let \(n_1<n_2<\cdots\) be an infinite sequence such that, for any choice of congruence classes \(a_i\pmod{n_i}\), the set of integers not satisfying any of the congruences \(a_i\pmod{n_i}\) has density \(0\). Is it true that for every \(\epsilon>0\) there exists some \(k\) such that, for every choice of congruence classes \(a_i\), the density of integers not satisfying any of the congruences \(a_i\pmod{n_i}\) for \(1\leq i\leq k\) is less than \(\epsilon\)? The answer is yes; the linked Lean proof formalizes Somani's argument.