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.
The first  
  convergents of the finite continued fraction  
\[[ a_1,a_2,... , \; a_n ]\]
\[C_1, \; C_2,..., \; C_n \]
. 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  
  is greater than the succeeding odd convergent  
. 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  
\[C_{2k}-C_{2k-1}=\frac{1}{q_{2k}q_{2k-1}} \le \frac{1}{(2k)(2k-1)} \]
\[q_k \ge k\]
  for large  
  the right hand side tends to zero, so the sequences tend to the same limit.

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