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How to prove that the complement in $\mathbb{R}$ of the set $A= \bigcup_{n \in \mathbb{N}} A_n$, where $A_n= (q_n-1/2^n,q_n+1/2^n)$ and $\{q_n\}_{n\in\mathbb{N}} = \mathbb{Q}$,
is uncountable ? How to know if it is at the first place ?

I am supposing there is no other way other than to suppose it is countable and find a contradiction ? so I am trying with $A^c= \{r_n\}_{n \in \mathbb{N}}$ making each $r_n$ not in the interval $(q_n-1/2^n,q_n+1/2^n)$ so this way we are supposing $\mathbb{R} = \bigcup_{n \in \mathbb{N}} ( (q_n-1/2^n,q_n+1/2^n) \cup {r_n} )$... how to find a real number not in here ??

Or maybe we find an interval inside this complement ?

In general though, what are the ways, or tools we can use to prove that some set is uncountable ?

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  • $\begingroup$ @Bernard: The question, however, is to prove that $\Bbb R\setminus A$ is uncountable. $\endgroup$ Oct 23, 2020 at 22:02
  • $\begingroup$ @BrianM.Scott: It seems I misread the question… (for my excuse, it's getting late here!) $\endgroup$
    – Bernard
    Oct 23, 2020 at 22:04
  • $\begingroup$ @Omar Edward Is a measure-theoretic approach ok with you? If so, see my answer below. And feel free to ask for clarification should it be needed. $\endgroup$ Oct 24, 2020 at 18:42

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Hints:

Denote by $\lambda: \mathcal B(\mathbb{R}) \to \mathbb{R} $ the Lebesgue measure on $\mathbb{R}$.

  1. Using $A = \cup_{n \in \mathbb{N}}A_n$ and $ \lambda(\cup_{n \in \mathbb{N}}A_n) \le \sum_n \lambda(A_n)$ can you find a bound on the Lebesgue measure of $A$?

  2. What is the Lebesgue measure of $A^c$? What is the Lebesgue measure of a countable set?


P.S. Keep in mind that this is a great example (another typical one is the fat Cantor set) of a closed, nowhere dense (why?) set with non-empty measure, which may look counter-intuitive at first!

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