It's so much more interesting if you happen to know about the "hydra game" (the Goodstein theorem), which is also a question about sequences and their terminati

It's so much more interesting if you happen to know about the "hydra game" (the Goodstein theorem), which is also a question about sequences and their termination at 0. https://en.wikipedia.org/wiki/Goodstein%27s_theorem https://en.wikipedia.org/wiki/Goodstein%27s_theorem But this is easily proved (using infinite ordinals, but it seems it could be proved just by coming up with a similar concept like the infinite ordinal arithmetic, basically providing an upper bound for each step of the algorithm and showing that there's an eventual maximum to these and then there's a monotonic decrease, and that the number of steps are always finite).