Disjunctive sequence

A disjunctive sequence over a finite alphabet $\Sigma$ is an infinite sequence in which each finite string over $\Sigma$ appears as a substring. For instance, the binary Champernowne sequence

$0\ 1\ 00\ 01\ 10\ 11\ 000\ 001\ldots$

formed by concatenating all finite, binary strings in lexicographic order, is clearly disjunctive.

Any normal sequence is disjunctive, but the converse is not true. For example, consider the disjunctive sequence

$0\ 0^{1}\ 1\ 0^{2}\ 00\ 0^{4}\ 01\ 0^{8}\ 10\ 0^{16}\ 11\ 0^{32}\ 000\ 0^{64}\ldots$

obtained by splicing exponentially long strings of 0's into a lexicographic ordering of all finite, binary strings. Most of this sequence consists of long runs of 0's, and so it is not normal.