You don't need strong induction to prove this. Consider the set of all numbers that cannot be expressed as a sum of Fibonacci numbers.
If this set were non-empty, it would have a smallest element $n_0$.
Now let $F_n$ be the largest Fibonacci number $< n_0$. Then $n_0 - F_n < n_0$ and thus $n_0 - F_n$ is a sum of Fibonacci numbers. Thus $n_0$ is also a sum of Fibonacci numbers. Contradiction.
Therefore there is no number that is not a sum of Fibonacci numbers.
Added: It is possible to prove that each $n \ge 2$ can be uniquely written as a sum of distinct Fibonacci numbers such that no two consecutive Fibonacci numbers appear in the sum. For example, $20 = 13 + 5 + 2$ and $200 = 144 + 55 + 1$ (Fibonacci Coding). Proof by strong induction.