Rational Number can be Expressed as Simple Finite Continued Fraction

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

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

Proof
Let $q = \dfrac a b$ 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 equations on the, we get:

This shows that $q$ has the SFCF $\sqbrk {q_1, q_2, q_3, \ldots, q_n}$.

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