# Proof of patterns in Fibonacci words

Let Fibonacci words over the alphabet $$\{0,1\}$$ be recursively defined by $$\omega_0=0$$, $$\omega_1=01$$, and $$\omega_n=\omega_{n-1}\omega_{n-2}$$ for $$n\geq{2}$$. I am trying to show that the patterns $$11$$ and $$000$$ never occur in any Fibonacci words. I am not sure how to go about proving this because I need to show that NO words contain these patterns. I tried using induction but I was not sure how to show that the statement is true for the next Fibonacci number.

1. All words (but $$w_0$$) start with $$01$$
2. No word ends with $$00$$