Rational Numbers and Simple Finite Continued Fractions are Equivalent

Theorem
Every simple finite continued fraction‎ has a rational value.

Conversely, every rational number can be expressed as a simple finite continued fraction‎.

Proof
Follows from
 * Finite Simple Continued Fraction has Rational Value
 * the proof below

Proof that Every Rational Number can be expressed as a SFCF
Let $\dfrac a b$ be a rational number expressed in canonical form.

That is $b > 0$ and $a \perp b = 1$.

By the Euclidean Algorithm, we have:

Thus from the system of eqns at the right hand side, we get:

This shows that $\dfrac a b$ has the SFCF $\left[{q_1, q_2, q_3, \ldots, q_n}\right]$.

Note
It can be seen from this proof that there is a close connection between continued fractions and the Euclidean Algorithm.