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.