Correspondence between Irrational Numbers and Simple Infinite Continued Fractions

Theorem
Let $\R\setminus \Q$ be the set of irrational numbers.

Let $S$ be the set of all simple infinite continued fractions in $\R$.

The mappings:
 * $\R\setminus \Q \to S$ that sends an irrational number to its continued fraction expansion
 * $S \to \R\setminus \Q$ that sends a simple infinite continued fractions to its value

are reverse bijections.

Proof
Note that indeed a Simple Infinite Continued Fraction Converges to Irrational Number.

The result follows from:
 * Continued Fraction Expansion of Irrational Number Converges to Number Itself
 * Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself

Also see

 * Correspondence between Rational Numbers and Simple Finite Continued Fractions