Properties of Convergents of Continued Fractions

Theorem
Let $F$ be a field, such as the field of real numbers $\R$.

Let $n \in \N \cup \{\infty\}$ be an extended natural number.

Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction in $F$ of length $n$.

Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators.

Let $C_0, C_1, C_2, \ldots$ be the convergents of $\left[{a_0, a_1, a_2, \ldots}\right]$.

Then the following results apply:

Simple continued fractions
Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a simple continued fraction in $\R$ of length $n$.