It is not.

The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.

This is usually described mathematically by saying that the representation of a decimal has to be countable, whereas the number 0.000...1 is not a countable representation.

I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.

Could you please explain a little more what "not a countable representation" means?

A countable set is one where you can reach any element in a finite amount of steps. See:

https://en.wikipedia.org/wiki/Countable_set

If you're saying we never get from the initial 0 to the trailing 1 in a finite number of steps, that's true and that's what DavidVoid is saying. But I haven't made the list of digits uncountably large by adding one element. Instead I put that element in a transfinite position in the ordering and I have to watch out for weird consequences.

I'm no expert, but countability doesn't depend on how the set is ordered. It depends on whether the elements can be placed in 1:1 correspondence with the integers. 1,0,0,... has a countable number of elements, and so does 0,0,0,...,1. They can be put in 1:1 correspondence with each other. This definition of countability is described in your link.

I meant to ask, does "countable representation" some kind of detailed definition that I can look at?

It's not the set of digits you use which is uncountable, it's the representation itself.

I think it's just another way to word DavidVoid's explanation a few comments up.

Ok. (a) I think you're saying that 0.00 ... 1 is not a real number. (b) Do we agree that lim n->inf 10^-n is a real number? (I.e. zero?) (c) Then you're saying that limit in "(b)" is not a reasonable definition of the string "0.00 ... 1". Is that correct?

That is correct.

The limit of lim n->inf 10^-n is exactly 0, it is not 0.00...1.

Since we have no common definition of what 0.00...1 might possibly mean, let's say we agree.

0.0000... is the repeating decimal representation of zero.