Positive Real Number has Simple Continued Fraction Expansion
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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