Doesn't $0.1,0.2,\ldots,0.9,0.11,0.12,0.13,\ldots,0.99,0.101,0.102,\ldots$ contain all values between $0.1$ and $1$, making that interval countable?

Question

Wouldn't a set of numbers that is ordered like

$$0.1,0.2,\ldots,0.9,0.11,0.12,0.13,\ldots,0.99,0.101,0.102,\ldots$$

(skipping values that repeat such as $0.10$, $0.100$, etc.)

necessarily include all values between $0.1$ and $1$ as a countable infinite? Since you could match each value after the decimal to a value on the countably infinite integer line.

Of course, I don't think that this is some "new discovery" or anything. I'm just trying to find a proof or example that demonstrates that this is still uncountably infinite.

Answer 1

This set does not include $\frac13$.

Answer 2

The decimals that terminate (i.e. finitely many nonzero digits) are countable, and this is what you show.

Note, however, that you miss lots of numbers, such as all the irrationals and even some rational like $1/3$ (more generally you miss every rational whose denominator is a multiple of some prime $p \neq 2,5$ when written in reduced form)