Doubt about Exercise 5, Chapter 1, Shakarchi [RA]

In $[0,1]$ let $C$ be a Cantor like set of positive measure and $E=C^{c}$. Then $m(O_n) \to m(\overline {E})\neq m(E)$ because $\overline {E}$ is the complement of the interior of $C$ and interior of $C$ is empty.

The idea is to make an assumption of Corollary 3.3 not satisfied. Let $\{x_n\}$ be an enumaration of $[0,1]\cap\mathbb{Q}$. Consider $$ E=\cup_{n=1}^{\infty} (x_n-\frac 1{4^n}, x_n+\frac1{4^n}).$$ This is an open set with Lebesgue measure at most $\sum 2/4^n = 2/3$.

Also, $E$ is dense in $[0,1]$. Then $m(O_n)\geq 1$ for each $n$.