<email@example.com> wrote in message news:firstname.lastname@example.org... > I see no limitations disallowing that interval between 0.999... and 1 > to be a metric space. It is infinitely sub-dividable and can be > understood to have its own metric. How this metric would relate to the > real numbers would require its own study. >
You would agree that R is a complete metric space, yet remove this portion? That my good sir is a contradiction.
> When I say decimal, I simply mean any number represent in our common > base ten positional numbering system. If this seems an overuse of them > word I would be interested in hearing a better simple description. >
Then, allowing for an infinite number of digits, lim_n->oo 1-10^n, or 0.9999... is expressible as a decimal.
You can write a proof of this fairly directly (go on :) ).