Explicitly give an infinity of rational numbers who come from below the number $9.123412341234 ...$, Explicitly give an infinity of rational numbers who come from below the number $9.123412341234 ...$, and an infinity who come from above.
I know this number is $\frac{91225}{9999}$, but how can I give that set of numbers?
Also I have to give an infinity of irational numbers who come from below the number $1.989898989898 ...$, and countless who come from above.
Any ideas or hint?

The set of rationals $\frac{n}{9999}$ where $n < 91225$ and $n$ is an integer is countably infinite with a cardinality of $ \aleph_0$. Likewise, the set of rationals $\frac{n}{9999}$ where $n > 91225$ and $n$ is an integer is countably infinite with a cardinality of $ \aleph_0$. In general, if you have a rational number $\frac{a}{b}$ there are countably infinite rational numbers $\frac{a-n}{b}$ and $\frac{a+n}{b}$ less than and greater than $\frac{a}{b}$ respectively, for rational $n$.
The second decimal expansion you give is equal to $\frac{197}{99}$. I leave it as an exercise to you to follow the procedure outlined above to produce your two infinite sets.
