Prove that in a bit string the string 01 occurs at most one

Prove that in a bit string, the string 01 occurs at most one more time than the string 10.

Solution

Lemma 1 P(w11w)P(w1w)P(w11w)P(w1w)

Lemma 2 P(w00w)P(w0w)P(w00w)P(w0w)

By iterating lemma 1 and 2 it suffices to prove this theorem only for the strings like 01010101, 10101, etc. (where no digit occurs twice in a row). The result is obviously true for these strings.