The short-cut isn’t necessarily wrong, multiplication by 10 in base 10 shifts the decimal point to the right. But (123 = 123.0), so we aren’t adding a zero, the zero was already there. That’s not the case for (0.\dot9) (if there are (L) decimal places, there can’t be any number in the (L+1)th place, because (L+1) is undefined).
I also suspect that we can prove the short-cut as a general theorem, since it’s the case that for any number in base x, multiplying by x shifts the decimal point one place to the right. (Tangent: is there a base-agnostic word for the ‘decimal’ point?)
I was thinking (10 * (3 * 0.\dot3) = (3 * \frac{10}{3}) = 9.\dot9), because (\frac{10}{3}) results in the same repeating decimals in the same way.
I agree this isn’t the construction we need to convince skeptics, but I’m not sure what construction the skeptics are using to define (0.\dot9). Again, without that construction, I’m not sure that we all mean the same thing when we say (0.\dot9).
Absolutely. Numbers aren’t closed on addition? A number where addition is defined for only half the number line (all negative numbers, but no positive numbers)? An end to all decimal expansions, such that there is uncertainty about what happens to any infinite expansion when multiplied by 10? Does that make a whole class of rational numbers that aren’t closed on multiplication? Is multiplication by (L) defined?
My impression is that the existence of (L) is ad hoc, unnecessary, and has a lot of unintended consequences. I don’t believe it can be proven from other standard axioms, so we’d be adding it as an additional axiom, and it isn’t clear why.
Ah, so the red herring is actually sport fishing! Fair enough.
The limit of the sequence is (0.0000… * \infty). My calculus is rusty, but I think that means the sequence doesn’t converge.