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.