Given:
$F_0 = 1 , F_1 = 1$, For 2 and so on $F_n = F_{n-1} + F_{n-2}$ (Fibonacci series).
Need to prove by induction:
For each natural $n$ there is a natural number $k$ and numbers:
$a_0, a_1, ..., a_k \in \{0,1\}$
So it exists: $$n = \sum_{i=0}^{k}a_iF_i$$