Bound for Difference of Irrational Number with Convergent

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Theorem

Let $x$ be an irrational number.

Let $\left \langle {C_n}\right \rangle$ be the sequence of convergents of the continued fraction expansion of $x$.


Then $\forall n \ge 1$:

$C_n < x < C_{n + 1}$ or $C_{n + 1} < x < C_n$
$\left|{x - C_n}\right| < \dfrac 1 {q_n q_{n + 1} }$


Proof

Immediate.

Note that:

$\left|{x - C_n}\right| < \left|{C_{n + 1} - C_n}\right| = \dfrac 1 {q_n q_{n + 1} }$

$\blacksquare$