I was reading this proof and needed a clarification. The question is too old to post in it, hence this question.
The original link is here: Outer Measure of the set $E=(\mathbb Q\times \mathbb R) \cup (\mathbb R\times \mathbb Q)$
The proof writer mentioned that using an enumeration $\mathbb{Q} = \{x_1, x_2, \cdots\}$, for each $\epsilon > 0$ let us consider the following sets:
$$ \begin{gathered} A_{i,n} = (x_i - \epsilon 4^{-i-n}, x_i + \epsilon 4^{-i-n}) \times (-2^n, 2^n), \\ B_{i,n} = (-2^n, 2^n) \times (x_i - \epsilon 4^{-i-n}, x_i + \epsilon 4^{-i-n}). \end{gathered} $$
It is easy to see that $E \subseteq \bigcup_{i,n\geq 1} A_{i,n} \cup B_{i,n}$.
I don't understand why $(-2^n, 2^n)$ would cover the real part of the set. How should we define $n$ to make this happen?