proof that a Zeckendorf representation represents a unique positive integer

Theorem. For any positive integer $n$, the Zeckendorf representation $Z$ (with $k$ elements all 0s or 1s) of $n$ is unique.

For our proof, we accept it as axiomatic that $F_0=F_1=1$ but $F_0$ is not used in the Zeckendorf representation of any number, and we also accept as axiomatic that all $F_i$ are distinct as long as $i>0$.

Proof.

Assume that there are two integers $a$ and $b$ such that , yet they both have the same Zeckendorf representation $Z$ with $k$ elements all 0s or 1s. We compute

$$c=\sum _{i=1}^k{Z}_i{F}_i,$$

where $F_x$ is the $x$th Fibonacci number. We can be assured that there is only one possible value for $c$ since all $F_i$ are distinct for $i>0$ and each $F_i$ was added only once or not at all, since each $Z_i$ is limited by definition to 0 or 1. Now $c$ holds the value of the Zeckendorf representation $Z$. If $c=a$, it follows that , but that would mean that $Z$ is not the Zeckendorf representation of $b$ after all, hence this results in a contradiction of our initial assumption. And if on the other hand $c=b$ and $c>a$, then this leads to a similar contradiction as to what is the Zeckendorf representation of $a$ really is.∎