Continued Fraction Expansion via Gauss Map

Theorem
Let:
 * $Y := \closedint 0 1 \setminus \Q$

that is, the irrationals between $0$ and $1$.

Let $T : Y \to Y$ be the Gauss map.

For $n \in \N_{>0}$, define $a_n : Y \to \N_{>0}$ be by:
 * $\map {a_n} x := \floor {\dfrac 1 {\map {T^{n - 1} } x} }$

where $\floor \cdot$ denotes the floor.

Let $x \in Y$.

Then $x$ has the simple infinite continued fraction: