Let $k\gg m$ mean that $k\ge m+2$. Show that every positive integer $n$ has a unique representation of the form $$n=F_{k_1}+F_{k_2}+ \dots +F_{k_r},$$ where $F_i$ is the $i$th Fibonacci number and $k_1\gg k_2\gg \dots \gg k_r \gg 0$.
Well I proved the existence part using induction on $n$. I am not getting anywhere in proving the uniqueness part. Any hints or solutions, please.
Show that $F_n$ is greater than any number you can express with an expression of your form using only $F_m$ with $m < n$. That is $$F_n>F_{k_1}+F_{k_2}+ \dots +F_{k_r},$$ where $n-1\ge k_1\gg k_2\gg \dots \gg k_r \gg 0$.
Then assume two distinct expressions for some number exist. Let $F_n$ be the largest number that occurs in the one, yet not in the other.
But by the observation at the start, we can never compensate this difference.