Continued Fraction Expansion of Irrational Number Converges to Number Itself

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Theorem

Let $x$ be an irrational number.


Then the continued fraction expansion of $x$ converges to $x$.


Proof

Let $(a_0, a_1, \ldots)$ be its continued fraction expansion.

Let $(p_n)_{n\geq 0}$ and $(q_n)_{n\geq 0}$ be its numerators and denominators.

Then $C_n = p_n/q_n$ is the $n$th convergent.

By Accuracy of Convergents of Continued Fraction Expansion of Irrational Number, for $n \ge 2$:

$\left|{x - \dfrac {p_n} {q_n}}\right| < \dfrac 1 {q_n q_{n + 1} }$

By Lower Bounds for Denominators of Simple Continued Fraction:

$q_nq_{n+1} \geq n$ for $n \geq 5$.

So from Basic Null Sequences and the Squeeze Theorem:

$\dfrac 1 {q_n q_{n+1} } \to 0$

as $n \to \infty$.

Thus $C_n = p_n/q_n$ converges to $x$.

That is, $(a_0, a_1, \ldots)$ converges to $x$.

$\blacksquare$


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