TITLE: Is the set of all rationals with denominators less than $10^6$ closed in $\mathbb{R}$?
QUESTION [5 upvotes]: I'm wondering if the set of all rationals with denominators less than $10^6$ is closed in the real number system. I think it is not, so that's what I've been trying to prove. I've tried looking at its complement and showing that it is open, but I didn't progress much. My last resort is to show that there is a sequence of rationals in my set converging to an irrational number (or to a rational whose denominator is greater than $10^6$). The problem is, how do I ensure that all simplified fractions in my sequence will have denominator less than $10^6$?

REPLY [10 votes]: Of course it's closed. Just blow it up by a factor of $(10^6)!$, and all its elements are integers.