Representation of Ternary Expansions

Theorem
Let $x \in \R$ be a real number.

Let $x$ be represented in base $3$ notation.

While it may be possible for $x$ to have two different such representations, for example:
 * $\dfrac 1 3 = 0.100000 \ldots_3 = 0.022222 \ldots_3$

it is not possible for $x$ be written in more than one way without using the digit $1$.

Proof
It is sufficient to show that two distinct representations represents two distinct numbers.

Let $a$ and $b$ two real numbers representable as the form above.

Their signs are easy to distinguish, so we consider $\size a$ and $\size b$.

There is some $n$ such that:
 * $\size a, \size b < 3^n$

In that case, $\dfrac {\size a} {3^n}$ can be represented as:
 * $0.a_1 a_2 a_3 \ldots$

and $\dfrac {\size b} {3^n}$ can be represented as:
 * $0.b_1 b_2 b_3 \ldots$

where $a_i, b_i$ are either $0$ or $2$.

Let $N$ be the smallest integer such that $a_N \ne b_N$.

assume that $a_N = 2$ and $b_N = 0$.

We have:

and thus $\size a$ and $\size b$ are distinct.