I think the way to formally prove this is to find the difference between the Fibonacci approximation and the usual conversion, and then to find whether that series is convergent or not. Someone who has taken the appropriate pre-calculus or calculus course could actually carry it out :P
However, I got curious about graphing it for distances “small enough” like from Earth to the sun (150 million km). Turns out, there’s always an error, but the error doesn’t seem to be growing. In other words, except for the first few terms, the Fibonacci approximation works!
This graph grabs each “Fibonacci mile” and converts it to kilometers either with the usual conversion or the Fibonacci-approximation conversion. I also plotted a straight line to see if the points deviated.