Positive Real Number has Simple Continued Fraction Expansion

Theorem

Let $x \in \R_{>0}$ be a (strictly) positive real number.

Then $x$ can be expressed as a simple continued fraction.

Proof

We have that $x$ is either rational or irrational.

$x$ rational

Let $x$ be rational.

Then from Rational Number can be Expressed as Simple Finite Continued Fraction, $x$ has a simple continued fraction expansion.

$x$ irrational

Let $x$ be irrational.

The result follows from Correspondence between Irrational Numbers and Simple Infinite Continued Fractions.

$\blacksquare$

Sources

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