Rational Numbers and Simple Finite Continued Fractions are Equivalent

Theorem

Simple finite continued fraction‎s and rational numbers are equivalent, in the following precise sense.

Simple Finite Continued Fraction has Rational Value

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

Let $\paren {a_0, \ldots, a_n}$ be a simple finite continued fraction‎ of length $n$.

Then its value $\sqbrk {a_0, \ldots, a_n}$ is a rational number.

Rational Number can be Expressed as Simple Finite Continued Fraction

Let $q \in \Q$ be a rational number.

Then $q$ can be expressed as a simple finite continued fraction.