Proof That Every Finite Continued Fraction is Rational
Theorem (Equivalence of Finite Continued Fractions and Rational Numbers)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.