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I am trying to improve my understanding and application of the Baire Category theorem and got stuck on the following problem;

Let $A$ be the set of real numbers whose decimal expansion does not use the digit 6. (A terminating decimal representation is used when it exists.) Show that there exists a real number which cannot be written as $a + q$ with $a ∈ A$ and $q ∈ \mathbb{Q}$.

My Idea was to write $A_n = A+\frac{1}{n}\mathbb{Z}$ so that the union of the $A_n$ is $A+\mathbb{Q}$. However it seems pretty hard to show that $A_n$ is nowhere dense, and I'm not really sure how to approach it.

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I think you can prove this as follows. Observe that $A$ is a nowhere dense set (just like the Cantor set). Set $A_{q} = \{a + q : a \in A \}$. Then $A_{q}$ is also nowhere dense, since $A_{q}$ is simply a translation of $A$. $B = \cup_{q \in Q} A_{q}$ is of first category, since it is a countable union of nowhere dense sets. But by the Baire category theorem $\mathbb{R}$ is of second category, so there must exist an $x \in \mathbb{R}$ that is not in $B$.

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  • $\begingroup$ Your $A_q$ definition is slightly off, but this seems legit. $\endgroup$ – Elie Bergman May 1 '18 at 15:05

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