Bound for Difference of Irrational Number with Convergent

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Theorem

Let $x$ be an irrational number.

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

Then $\forall n \ge 1$:

$C_n < x < C_{n + 1}$ or $C_{n + 1} < x < C_n$

$\size {x - C_n} < \dfrac 1 {q_n q_{n + 1} }$

Proof

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Immediate.

Note that:

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

$\blacksquare$