Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$.
For example: $32 = 21 + 8+3 = F_8+F_6+F_4$
Research effort:
The base step it's simple:
Let $k=1$ it can be britten as $k=1=F_2$
For the inductive step I considered:
Let $k = k F_2 =F_2+F_2+\cdots+F_2 = \sum_{i=2}^w a_iF_i, a_i =\{0,1\} $
Then if $k$ suffice I want to see if $k+1$ suffices too... But I'm not realy seeing how to use the inductive hypothesis, so I assume It's wrong.
Any thoughts on which can the inductive step be?