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