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.

Edit: Here’s another graph

So it turns out:

• Fibonacci-approximated kilometers are always higher than the usual-conversion kilometers
• At most, the difference between both is 25%. That happens early on in the terms.
• After that, the percentage difference oscillates around a value and comes closer to it.
• When talking about more than 100 miles, the percentage change approximates 0.54.

TL;DR:

• Yes, the Fibonacci trick is true forever as you go higher in the sequence if you’re willing to accept a 0.54% error.