# Irrational Number is Limit of Unique Simple Infinite Continued Fraction

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## Theorem

Let $x$ be an irrational number.

Then the continued fraction expansion of $x$ is the unique simple infinite continued fraction that converges to $x$.

## Proof

Follows from:

- Continued Fraction Expansion of Irrational Number Converges to Number Itself
- Simple Infinite Continued Fraction is Uniquely Determined by Limit.

$\blacksquare$

## Also see

- Correspondence between Irrational Numbers and Simple Infinite Continued Fractions, a more precise statement