Real Numbers are Uncountably Infinite/Set-Theoretical Approach: Proof 2

Theorem
The set of real numbers $\R$ is uncountably infinite.

Proof
By Cantor's Theorem there is no surjection:
 * $\N \twoheadrightarrow \mathcal P \left({\N}\right)$

Additionally, we have [[Power Set of Natural Numbers Not Countable].

Therefore, if we can show that $\mathcal P \left({\N}\right)$ injects into $\R$ then there is no injection $\R \hookrightarrow \N$ and $\R$ is uncountable.

To prove the theorem we construct an injection, say $f$.

For a subset $S \subseteq \N$, let $\chi_S$ be the characteristic function of $S$, and let $d_i = \chi_S \left({i}\right)$ for all $i \in \N$.

If the sequence $\left\langle{d_i}\right\rangle_{i \in \N}$ does not terminate in an infinite sequence of $1$'s, then $f$ sends $S \subseteq \N$ to the binary expansion of a number in $\left[{0 \,.\,.\, 1}\right)$:


 * $0.d_1 d_2 d_3 d_4 \ldots$

If the sequence $\left\langle{d_i}\right\rangle_{i \in \N}$ does terminate in an infinite sequence of $1$'s, then $f$ sends $S$ to the binary integer:


 * $1 d_1 d_2 d_3 \ldots d_k.000 \ldots \ge 1$

where $d_k$ is the last member of the sequence not equal to $1$.

Injectivity of $f$ follows from the uniqueness statement of Existence of Base-N Representation.