Equivalence of Definitions of Cantor Set

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

Let $x \in \closedint 0 1$.

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

First we note that from Sum of Infinite Geometric Sequence:
 * $\displaystyle 1 = \sum_{n \mathop = 0}^\infty \frac 2 3 \paren {\frac 1 3}^n$

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 $3$.