Erdős Problem #25 — Existence of logarithmic density after excluding residues

Let $n_1 < n_2 < \ldots$ be an arbitrary sequence of integers. Now the following question is perhaps of interest: Exclude one or several residues mod $n_i$ (where only the integers $\geq n_i$ are excluded). Is it true that the logarithmic density of the integers which are not excluded always exists? This question seems difficult even if we only exclude one residue mod $n_i$ for every $n_i$.