I know of proofs that the rationals are not $G_{\delta}$, so I was just wondering what the following set is equal to: $$ \bigcap_m \bigcup_{r_n\in \Bbb Q}(r_n-2^{-n}/m,r_n+2^{-n}/m) $$ I know it contains $\Bbb Q$ but has measure 0 so clearly it is not the whole real line. However, it is also $G_{\delta}$ so it must contain some irrationals. Is there a way to show that this set contains an irrational number that does not use the fact that $\Bbb Q$ is not $G_{\delta}$? Perhaps even a way to find/construct such an irrational explicitly?
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Take a look at the answer by Robert Israel in the thread this is a duplicate of. – Andrés E. Caicedo Oct 19 '18 at 03:26
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It's very, very difficult to be explicit without you being explicit about how you've enumerated $\mathbb{Q}$! By changing $(r_n)$, you will change the intersection. – Theo Bendit Oct 19 '18 at 03:30