Can conclude $\mu\left(A\right)=0$ and why?

Question

Let $\left\{ a_{n}\right\} $ be a sequence of real numbers. Let $A$ be the set of all real numbers $x$ belonging to infinite open intervals $\left(a_{n},a_{n}+2^{-n}\right)$ . Can conclude $\mu\left(A\right)=0$ and why? ($\mu$ is Lebesgue measure).

Thanks in advanced.

Answer 1

Are you familiar with the following result?

Theorem If $\{E_n\}$ is a sequence of measurable subsets of $\mathbb{R}$ such that $m(E_1)<\infty$ and $E=\bigcap_{n\in\mathbb{N}} E_n$, then $m(E)=\lim_{n\to\infty} m(E_n)$.

Answer 2

Well $A \subseteq (a_n,a_n+\frac{1}{2^n})$ and

$\mu (a_n,a_n+\frac{1}{2^n})=\frac{1}{2^n}$ so $\mu(A) \leq \frac{1}{2^n}$ for all $n$ thus $\mu(A)=0$