Real Numbers are Uncountably Infinite/Proof 2 using Ternary Notation

Proof
Define a mapping $f: \mathcal P \left({\N_{>0}}\right) \to \R$ thus:


 * $f \left({S}\right) = 0.d_1 d_2 \ldots$, interpreted as a ternary expansion where $\left\langle{d_n}\right\rangle$ is the characteristic function of $S$.

That is:
 * $\displaystyle f \left({S}\right) = \sum_{i \mathop \in S} 3^{-i}$

By the lemma, $f$ is an injection.

Suppose for the sake of contradiction that $\R$ is countable.

Then there is an injection $g: \R \to \N$.

By Composite of Injections is Injection, $g \circ f: \mathcal P \left({\N}\right) \to \N$ is an injection.

But this contradicts No Injection from Power Set to Set.

Thus $\R$ is not countable.