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
this needs Correspondence between Irrational Numbers and Simple Infinite Continued Fractions
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Immediate.
Note that:
- $\left|{x - C_n}\right| < \left|{C_{n + 1} - C_n}\right| = \dfrac 1 {q_n q_{n + 1} }$
$\blacksquare$
