# How to prove this set is uncountable?

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 ?

• @Bernard: The question, however, is to prove that $\Bbb R\setminus A$ is uncountable. Oct 23, 2020 at 22:02
• @BrianM.Scott: It seems I misread the question… (for my excuse, it's getting late here!) Oct 23, 2020 at 22:04
• @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. Oct 24, 2020 at 18:42

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?