Equivalence of Definitions of Cantor Set

Theorem
The definitions of the Cantor set $\mathcal C$:


 * $(1): \quad$ As a limit of intersections
 * $(2): \quad$ Via ternary representations
 * $(3): \quad$ As a limit of a decreasing sequence

are logically equivalent.

Proof
Let $\mathcal C_n$ be defined as in $(1)$.

Let $x \in \left[{0 .. 1}\right]$.

We need to show that:
 * $x$ can be written in base $3$ without using the digit $1 \iff \forall n \in \Z, n \ge 1: x \in C_n$

First we note that from Sum of Infinite Geometric Progression:
 * $\displaystyle 1 = \sum_{n=0}^\infty \frac 2 3 \left({\frac 1 3}\right)^n$

... that is:
 * $1 = 0.2222 \ldots_3$

Thus any real number which, expressed in base $3$, ends in $\ldots 10000 \ldots$ can be expressed as one ending in $\ldots 02222 \ldots$ by dividing the above by an appropriate power of $3$.