For $x\in [0,1), \ \ x=\sum_{n=1}^{\infty} \frac{a_nx}{10^n}$ where $a_i$ is a digit in 0,1,...,9, define $I(a_1, \cdots ,a_n)$ be all $y\in [0,1)$ such that the decimal expansion of $y$ has first $n$ digit equal to $a_1, \cdots ,a_n$, respectively. For subset $A\subset [0,1)$, let $N_n(A)=\text{card }\left\{ I(a_1, \cdots, a_n)|I(a_1, \cdots, a_n)\cap A\neq \emptyset \right\}$.
Prove that there exists a Borel set $A\subset [0,1)$, $$m(A)< \lim_{n\to \infty} \frac{N_n(A)}{10^n}$$ where $m$ is Lebesgue measure.
I have proved that $m(A)\le \lim_{n\to \infty} \frac{N_n(A)}{10^n}$ for all Borel set $A$, but can't find a case this inequality is strict. Thanks for any help.