Continued Fraction Expansion of Irrational Number Converges to Number Itself

Theorem

Let $x$ be an irrational number.

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

Proof

Let $\sequence {a_0, a_1, \ldots}$ be its continued fraction expansion.

Let $\sequence {p_n}_{n \mathop \ge 0}$ and $\sequence {q_n}_{n \mathop \ge 0}$ 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 $n \ge 2$:

$\size {x - \dfrac {p_n} {q_n} } < \dfrac 1 {q_n q_{n + 1} }$

By Lower Bounds for Denominators of Simple Continued Fraction:

$q_n q_{n + 1} \ge n$ for $n \ge 5$

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 $x$.

$\blacksquare$

Also see

• Irrational Number is Limit of Unique Simple Infinite Continued Fraction