An infinite sequence is normal if all strings of equal length occur with equal asymptotic frequency.

Formally, let $\Sigma$ be a finite alphabet of $b$ digits. Let $S$ be an infinite sequence and $\omega$ a finite sequence, both over $\Sigma$, i.e., $S\in\Sigma^\infty$ and $\omega\in\Sigma^*$. Define $N_S(\omega, n)$ to be the number of times the string $\omega$ appears as a substring in the first $n$ digits of the sequence $S$. The sequence $S$ is normal if for all finite strings $\omega\in\Sigma^*$,

$$ \lim_{n\rightarrow\infty}{N_S(\omega,n)\over n}={1 \over b^{\left|\omega\right|}} $$

Or you can find the definition here: http://en.wikipedia.org/wiki/Normal_number

The webpage in wiki above also states a property of normal sequences: "A sequence is normal if and only if every block of equal length appears with equal frequency. (A block of length $k$ is a substring of length $k$ appearing at a position in the sequence that is a multiple of k: e.g. the first length-$k$ block in $S$ is $S[1..k]$, the second length-$k$ block is $S[k+1..2k]$, etc.)"

Wiki gives me the reference. More specifically, it says this result was made explicit in the work of Bourke, Hitchcock, and Vinodchandran (2005). But I cannot figure out which theorems in the reference imply this property and how they do.

This "block characterization of normality" seems a natural property and should be easy to prove, but so far I still have difficulties in proving the "only if" direction. On the other hand this result seems important as it implies several other properties (see wiki). One paper I read about the connections between normal sequences and finite automata also relies on this result.

So I will be grateful if someone can help.