A simple question.

[Zenith]

0.00...0001 is finitely long because it ends with a 1 (i.e. there is nothing after the last 1). 0.111... is infinitely long because it doesn't have an end by definition.

By the same logic I can say that 0.(1) = 0.11...111 is finite because it ends with "1". In other words, my example hasn't an end by definition, either. If this "0.00...001" does not look as you'd like, then use this representation:

"1/10 * 1/100 * 1/1000 ..." - because this is clearly not an "invalid" thing.

As I've said before, using an infinite geometric series doesn't use infinity in the actual calculation.

I don't care if infinite geometric series use or not the infinity in the actual calculation. I also don't care how this thing should be called. But, if you insist that "1/10 * 1/100 * 1/1000 ..." must not be used, perhaps you can show where it's written that this thing is forbidden to be used.