We call a pair of codes x and y to be a losing pair if the Fibonacci ex- pansion for one of them ends in an even number of zeros and Fibonacci expansion for the other is obtained by adding a zero to the right of the expansion for the first one. Show that given any positive integer N, there is a losing pair x and y such that x − y = N.
The following proof below is what I found. Is there another way to write it more simpler and easier to explain.
Sure. Let $\sum_i a_i F_i$ be the F-expansion for $N$. Then \begin{align} N &=\sum a_i F_i \\&=\sum a_i (F_{i+2}-F_{i+1}) \\&=\left(\sum a_i F_{i+2}\right)-\left(\sum a_i F_{i+1}\right) \\&=\underbrace{\left(\sum a_{i-2} F_{i}\right)}_x-\underbrace{\left(\sum a_{i-1} F_{i}\right)}_y. \end{align} The expressions for $x$ and $y$ are their F-expansions, and you can see that they are a losing pair.