Using induction for

.

----

Each of the integers

is a sum of numbers from the set

none of which are repeated. Select

such as,

But because,

We can express,

as sum of numbers from

none of which are repeated. Thus, as a result,

is expressable as a sum of the numbers,

without repetitions.

This means that any of the numbers,

is expressable from the set,

and the induction is complete.

Thus any number is expressable as a sum of distinct fibonacci numbers.

Note if any of the two fibonacci numbers are consecutive then they can be combined to give the number fibonacci number. Countinuing combining adjacent fibonacci number you have proven

Zeckendorf's Representation