Proof That Infinite Continued Fractions are Irrational
Any infinite Continued Fraction is irrational.
Proof
Let
\[x =[ a_1, \; a_2,..., ]\]
be an infinite continued fraction. For each \[n \gt 1\]
\[C_n = \frac{p_n}{q_n}, \; C_{n+1} = \frac{p_{n+1}}{q_{n+1}}\]
By the Relative Size of Convergents of Finite Continued Fractions ,
\[C_{2k+1} \lt x \lt C_{2k}\]
, so \[\| x- C_n \| \lt \| C_n- C_{n+1} \| = \frac{1}{q_nq_{n+1}}, by the Properties of Convergents of Finite Continued Fractions.\]
.Suppose
\[x\]
is rational, say \[x= \frac{r}{s}\]
. Then \[\| x - C)n \| = \| \frac{r}{s} - \frac{p_n}{q_n} \| = \| \frac{rq_n - sp_n}{sq_n} \| \lt \frac{1}{q_nq_{n+1}}\]
.Then
\[\| rq_n - s p_n \| \lt \frac{1}{q_{n+1}} \lt 1\]
, which is impossible. 
