Cantor Set is Uncountable/Proof 2

Theorem
The Cantor set $\mathcal C$ is uncountable.

Proof
It follows from Representation of Ternary Expansions that every string of the form $0.nnnnn \ldots$ where $n \in \left\{{0, 2}\right\}$ is an element of $\mathcal C$.

We also have that every string of the form $0.nnnnn \ldots$ where $n \in \left\{{0, 1}\right\}$ is an element of $\left[{0 \,.\,.\, 1}\right] \subset \R$ expressed in binary notation.

Let $f: \mathcal C \to \left[{0 \,.\,.\, 1}\right]$ be the function defined by:
 * $\forall x \in \mathcal C: f \left({x}\right) = \text{ the number obtained by replacing every } 2 \text { in } x \text { with a } 1$

where $x$ is expressed in base $3$ notation.

It is clear from the above that $f$ is a surjection.