Equivalence of Definitions of Convergent of Continued Fraction

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

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

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

Let $k \leq n$ be a natural number.

Proof
This follows immediately from Value of Finite Continued Fraction equals Numerator Divided by Denominator.