> Represent each real number x∈[0,1) as a sequence of DIGITS, where each DIGIT is either in {0,…,8} or is of the form 10∗(10k−1)+i with k≥1 and i∈{0,…,8} (i.e., in {90,…,98;990,…,998;9990,…,9998;…}.
He's iterating to infinity, that sure as shit sounds countable to me.
Anyway, the conclusion that I've come to is that you can map R onto a subset of R^n such that for all values {r[1] ... r[n]} in R^n there is a value in the R -> R^n mapping within all finite bounds.
So as long as you stay away from infinitesimals (not all of them), the domains are equivalent.
https://en.wikipedia.org/wiki/Infinitesimal
> In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero.
But as far as I can tell, the reverse mapping has an R^(n - 1) orthogonal that is completely unaccounted for, within those finite bounds.
(And breaking infinitesimals might cause some trouble for calculus.)