Equivalence of Definitions of Convergent of Continued Fraction

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

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

Let $C = \sqbrk {a_0, a_1, a_2, \dotsc}$ be a continued fraction in $F$ of length $n$.

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

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