Accuracy of Convergents of Convergent Simple Infinite Continued Fraction

Theorem
Let $C = \tuple {a_0, a_1, \ldots}$ be an infinite simple continued fraction in $\R$.

Let $C$ converge to $x \in \R$.

For $n \ge 0$, let $C_n = \dfrac {p_n} {q_n}$ be the $n$th convergent of $C$, where $p_n$ and $q_n$ are the $n$th numerator and denominator.

Then for all $n \ge 0$:
 * $\size {x - \dfrac {p_n} {q_n} } < \dfrac 1 {q_n q_{n + 1} }$.

Proof
We show that either:
 * $x \in \closedint {C_n} {C_{n + 1} }$

or:
 * $x \in \closedint {C_{n + 1} } {C_n}$

so that the result follows from:
 * Difference between Adjacent Convergents of Simple Continued Fraction
 * Distance between Point of Real Interval and Endpoint is at most Length

Odd case
Let $n \ge 1$ be odd.

By Limit of Subsequence equals Limit of Sequence:
 * $x = \ds \lim_{k \mathop \to \infty} C_{2 k}$

For all $2 k \ge n + 1$, by:
 * Even Convergents of Simple Continued Fraction are Strictly Increasing
 * Even Convergent of Simple Continued Fraction is Strictly Smaller than Odd Convergent

we have:
 * $C_{n + 1} \le C_{2 k} < C_n$

By Lower and Upper Bounds for Sequences:
 * $x \in \closedint {C_{n + 1} } {C_n}$

Even case
Let $n \ge 0$ be even.

By Limit of Subsequence equals Limit of Sequence:
 * $x = \ds \lim_{k \mathop \to \infty} C_{2 k + 1}$

For all $2 k + 1 \ge n + 1$, by:
 * Odd Convergents of Simple Continued Fraction are Strictly Decreasing
 * Even Convergent of Simple Continued Fraction is Strictly Smaller than Odd Convergent

we have:
 * $C_n < C_{2 k + 1} \le C_{n + 1}$

By Lower and Upper Bounds for Sequences:
 * $x \in \closedint {C_n} {C_{n + 1} }$

Also see

 * Accuracy of Convergents of Continued Fraction