A simple question.

[Tiberius]

By the same logic I can say that 0.(1) = 0.11...111 is finite because it ends with "1".

A few things wrong with this statement:

1) 0.11...111 would be finite even if it didn't end with a 1, since I can name a number higher than it (0.2), and hence it is not infinite. What you mean is that it is finitely long, which is perfectly true if it ends with a 1, and the way you have written it (0.11...111) shows this.

2) 0.(1) is not the same as 0.11...111, since by definition 0.(1) doesn't have an end, and 0.11...111 has a very visible end. The correct notation for 0.(1) is 0.111... (the ellipsis indicating that the 1 repeats endlessly).

3) Whilst 0.11...111 does end with a very visible 1, 0.(1) or 0.111... does not. The ellipsis makes it endless, and so whilst it is a finite value, it is not of finite length.

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.

Your example does have an end: 0.00...001 <-- there it is.

Whilst your calculation is a valid one, the answer to it diverges to 0 as I pointed out, and as such it is not a valid answer to the question "what is the first number that comes after 0?" because either (a) it eventually becomes 0, or (b) it diverges to a value which is impossible to calculate (since we would need a infinite amount of time in order to do it).

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.

I never insisted that your calculation must not be used, I said that the calculation diverges to 0, which in mathematics either means it becomes equal to 0, or becomes infinitely small and cannot be calculated.

I asked you what the first number to come after 0 was, and you have not given me a valid answer.