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    $\begingroup$ Excellent example! It took me a minute to fill in the reasoning, so I'll write it here for the benefit of future me. Each $V_n$ is co-meager since you've removed an interval around each rational, and a countable intersection of co-meager sets is co-meager. A co-meager set can't be meager or else all of $\mathbb R$ would be meager, which is isn't. $\endgroup$
    – Anton Geraschenko
    Commented Oct 25, 2010 at 5:37
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    $\begingroup$ It is a common error for a beginner to do this construction, then to think that $\cap_n V_n = \mathbb{Q}$. And then what is the instructor to do? Does the instructor have to cite something as esoteric as the Baire Category Theorem to disabuse that beginner of the erroneous idea? $\endgroup$
    – Gerald Edgar
    Commented Oct 26, 2010 at 14:42
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    $\begingroup$ @Gerald Edgar: good question, maybe argue that we can make a real number belong to each $V_n$ by specifying more and more of its binary expansion. We can even take breaks, i.e., after we have made $r$ start as $0.r_0\cdots r_{k_1}$ and thereby ensured $r\in V_1$ we can append $r_{k_1+1}\cdots r_{\ell}$ so as to make sure $r\ne p_1$. Then we add $r_{\ell+1}\cdots r_{k_2}$ to make $r\in V_2$ and so on. This kind of thing also shows $\cap_n V_n$ is size continuum. $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Oct 26, 2010 at 18:39
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    $\begingroup$ @GeraldEdgar, after you present the Bjørn Kjos-Hanssen's argument to dispel students myth, you turn around to students and say "Now students, this type of topological diagonal argument technique can be used to prove existence of a lot of things. Mathematicians have encapsulated this technique into a theorem and it's called Baire Category Theorem." That might make students less intimidated by Baire Category Theorem. $\endgroup$
    – Jisang Yoo
    Commented Dec 2, 2013 at 18:19