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.