Bound for Difference of Irrational Number with Convergent
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this needs Correspondence between Irrational Numbers and Simple Infinite Continued Fractions
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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
this needs Correspondence between Irrational Numbers and Simple Infinite Continued Fractions
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Immediate.
Note that:
- $\size {x - C_n} < \size {C_{n + 1} - C_n} = \dfrac 1 {q_n q_{n + 1} }$
$\blacksquare$