Equivalent Characterisations of Irrational Periodic Continued Fraction

Theorem
Let $x \in \R \setminus \Q$ be an irrational number.

Let $\sequence {a_n}_{n \mathop \ge 0}$ be its continued fraction.

Let $N \ge 0$ be a natural number.


 * $(1):\quad$ The sequence of partial quotients $\sequence {a_n}_{n \mathop \ge 0}$ is periodic for $n \ge N$.
 * $(2):\quad$ The sequence of complete quotients $\sequence {x_n}_{n \mathop \ge 0}$ is periodic for $n \ge N$.
 * $(3):\quad$ There exists $M > N$ such that $x_M = x_N$.