Irrational Number is Limit of Unique Simple Infinite Continued Fraction

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.

Also see

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