The Fibonacci sequence $f_1, f_2, f_3, \ldots$ is defined by $f_1 = 1, f_2 = 2$, and $f_m = f_{m−1} + f_{m−2}$ for each integer $m \ge 3$. Prove that every $n \in \mathbb{N}$ can be expressed as the sum of distinct integers from the Fibonacci sequence.

Note: You may use without proof that the Fibonacci sequence is an increasing sequence

I am stuck in this question. Are we supposed to use strong induction here? I am trying it but get stuck. Any help would be appreciated!



Use strong induction, assuming this is true up to some $n-1 \in \mathbb{N}$. Then, for $n$, find the largest Fibonacci member $f$ below $n$ and apply strong induction to $n - f$.


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