Representations for 1 in Golden Mean Number System

Theorem
Then there are infinitely many ways to express the number $1$ in the golden mean number system.

Proof
We have that:
 * $\phi^0 = 1$

and so $1$ has the representation $S_1$ as:
 * $S_1 := \left[{1 \cdotp 00}\right]_\phi$

By inspection it is seen that $S_1$ is the simplest form of $1$.

Expanding $S_1$:
 * $S_2 := \left[{0 \cdotp 11}\right]_\phi$

which can then be expressed as:
 * $S_2 := \left[{0 \cdotp 1100}\right]_\phi$

Expanding $S_2$:
 * $S_3 := \left[{0 \cdotp 1011}\right]_\phi$

and so:
 * $S_4 := \left[{0 \cdotp 101011}\right]_\phi$

Continuing in this way we obtain an infinite sequence $\left\langle{S_n}\right\rangle$ all of which are a representation of $1$.