Another way to think about it: forget about how many people there are in total, I can observe x blue eyes. Let m be the minimum globally known number of people with blue eyes. After the announcement m = 1.
On that day (and any subsequent) if your x < m you know you have blue eyes. If at the end of the day noone leaves, m must be at least m + 1.
On that day (and any subsequent) if your x < m you know you have blue eyes. If at the end of the day noone leaves, m must be at least m + 1.