But this is not a compelling argument to somebody in this situation. While correct, it feels identical to saying "it just is".

The whole numerical representation scheme really is just a man made system. If you dig deep enough the veneer disappears. This is especially noticeable when you start to see things like numbers that are finite in decimal but have infinite repeating patterns binary.

Though you could treat it as an infinite series in a similar way:

0.9 -> 0.99 -> 0.999 -> ... -> ?

0.1 -> 0.01 -> 0.001 -> ... -> ?

The first is an infinite series:

  9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...

The second is not?!

I mean you could if you want to:

1/10 + -9/100 + -9/1000 + -9/10000 + ...

But I just meant them as series, not necessarily as sums.

Or

  1 - 9/10 - 9/100 - ...

or

  1 - ( 9/10 + 9/100 + ... )

So it becomes circular:

  0.00...1 = 1 - 0.999...

(In math a series is a sum:

https://en.m.wikipedia.org/wiki/Series_(mathematics)

Woops, you're totally right, I think I should've said "sequence".