As I said when you posted this question on Mathoverflow, I would still like to hear your motivation for asking this question, and I don't think it should receive a full answer until you've given one. That's why the question has a negative score on the other site.

It would also be good to hear what approaches you've tried toward investigating the problem more systematically. If you're completely stuck, Paul's comments about separately considering those open sets which do and don't contain $a_0$ are a good hint to start you off.

If that doesn't get you anywhere either, I'll consider posting a full solution to the problem if you first share your motivation for asking about it.