I'm not so well up on my math, but I was wondering if the set of "[EDIT: finitely] decimal expressible" real numbers $R$ between $[0,1]$ is countable?
Wikipedia gives the definition:
A set $S$ is called countable if there exists an injective function $f$ from $S$ to the natural numbers $\mathbb{N}$
It would seem to my mathematically naïve brain that such an injective function exists—for any element $r \in R$ you can simply "reverse the number" and remove the decimal point to end up with, e.g.,:
$f(0.0001) = 10000$
This leads me to three questions:
- Is this intuition correct?
- Would this constitute a proof?
- Can the result be extended to general real numbers between $[0,1]$?
Any help with terminology would also be great... E.g., I don't know a mathematical function for "reversing" a number...
Thanks!
(P.S., I've just noted a specific question here which is related and gives the general principle, but still doesn't answer my specific examples.)