# Bound for Difference of Irrational Number with Convergent

## 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:

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

$\blacksquare$