## Convergence of Infinite Continued Fractions

Theorem (Convergence of the Convergents of Infinite Continued Fractions)
An infinite continued fraction is any continued fraction of the form
$[ a_1,a_2,... ]$
. For such a continued fraction, the Convergents of Continued Fractions
$C_n, \; n=1, \; 2, \; 3,...$
tend to a limit.
Proof
The first
$n$
convergents of the finite continued fraction
$[ a_1,a_2,... , \; a_n ]$
are
$C_1, \; C_2,..., \; C_n$
. Each convergent
$C_k = \frac{p_k}{q_k}$
is defined by This Theorem.
By the Relative Size of Convergents of Finite Continued Fractions of Convergents, the even convergents form a decreasing sequence, the odd convergents form an increasing sequence, and each even convergent
$C_{2k}$
is greater than the succeeding odd convergent
$C_{2k+1}$
. The sequence of even convergents are bounded below by the odd convergents, and the sequence of odd convergents are bounded above by the even convergents, so each sequence converges to a limit.
From the Properties of Convergents of Finite Continued Fractions, we get
$C_{2k}-C_{2k-1}=\frac{1}{q_{2k}q_{2k-1}} \le \frac{1}{(2k)(2k-1)}$
.
since
$q_k \ge k$
for large
$k$
.
As

the right hand side tends to zero, so the sequences tend to the same limit. 