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.

$\blacksquare$

Also see

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

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continued fraction