Convergents of Simple Continued Fraction are Rationals in Canonical Form

Theorem
For all $k \ge 1$, $\dfrac {p_k} {q_k}$ is in canonical form:
 * $p_k$ and $q_k$ are coprime
 * $q_k > 0$.

Proof
Let $d = \gcd \left\{{p_k, q_k}\right\}$.

From Common Divisor Divides Integer Combination:
 * $p_k q_{k-1} - p_{k-1} q_k$ is a multiple of $d$.

From Difference between Adjacent Convergents of Simple Continued Fraction:
 * $d \mathrel \backslash \left({-1}\right)^k$

where $\backslash$ denotes divisibility.

It follows that:
 * $d = 1$

Note that:
 * $q_1 = 1$

and:
 * $\forall k \ge 2: a_k > 0$

It follows that $q_k > 0$ for all $k \ge 1$ from definition of denominator.

Hence the result.