## Proof That Every Finite Continued Fraction is Rational

Every simple finite continued fraction is rational and vice versa.

Proof is by induction. Let

\[P(k)\]

be the statement that the finite continued fraction of length \[k\]

is rational. Obviously \[P(1)\]

is true since \[[ a_1 ] = a_1\]

is rational for \[a_1\]

an integer.Suppose then that

\[P(k)\]

is true, so that every continued fraction of length \[k\]

is rational.By The First Continued Fraction Identity,

\[[ a_1,a_2,...,a_k,a_{l+1} ] =a_1+ \frac{1}{[ a_2,...,a_k,a_{l+1} ]}\]

and \[\frac{1}{[ a_2,...,a_k,a_{l+1} ]}\]

, being a finite continued fraction of length \[k\]

, is rational, so \[P(k+1)\]

is true.