Let $E$ be the set of all $x \in [0,1]$ with the following property:
If $0.x_1x_2x_3...$ is a decimal expansion of $x$, then for each integer $k \geq 1$, the string $x_1x_2x_3...x_k$ appears infinitely often in the sequence $x_1x_2x_3...$ of the decimal expansion of $x$.
Prove that the set $[0,1]\setminus E$ has Lebesgue measure zero.
Now I know that if I fix a word $w=x_1x_2x_3...x_k$ of length $k$, $w\in\ \{0,1\}^k$; the frequency of $x$ appearing in the decimal expansion for almost every $x\in \{\frac{p}{10^n}| 0<p\leq 10^n;n\geq 1\}$ is $\frac{1}{10^k}$ then measure of $E$ would be $\sum_\limits{k=1}^\infty \frac {1}{10^k}=1 $. So, I am done right?