This is not true. For instance, sequence 00 does not occur in sequence 010110111011110111110111111... which can hardly be described as "repeating" (or period

This is not true. For instance, sequence 00 does not occur in sequence 010110111011110111110111111... which can hardly be described as "repeating" (or periodic) sequence. The point here is normality. Fix alphabet A having b letters. We say that a sequence S of elements of A is normal if, for every word w, we have lim n->inf C(S, w, n)/n = b^(-k) where C(S, w, n) is the number of occurrences of w in first n letters of S, and k is length of w. More intuitively, S is normal if every word occurs in S with the same asymptotic frequency. So, if a sequence does not contain some word as a substring, it cannot be normal by above definition.