Simple Infinite Continued Fraction Converges

Theorem
The value of $\left[{a_1, a_2, a_3, \ldots}\right]$ is the limit of the sequence of convergents $\left \langle {C_n}\right \rangle$.

Proof
We need to show that for any SICF its sequence of convergents $\left \langle {C_n}\right \rangle$ always tends to a limit.

Several techniques can be used here, but a quick and easy one is to show that $\left \langle {C_n}\right \rangle$ is a Cauchy sequence.

By Difference between Adjacent Convergents of Simple Continued Fraction:
 * $\left|{C_{k + 1} - C_k}\right| = \dfrac 1 {q_{k + 1} q_k}$

From Denominators of Simple Continued Fraction are Strictly Increasing:
 * $q_k > k$

for sufficiently large $k$, $q_k > k$.

So:
 * $\dfrac 1 {q_{k + 1} q_k} < \dfrac 1 {\left({k + 1}\right) k} < \frac 1 {k^2}$

But $\left \langle {\dfrac 1 {k^2} }\right \rangle$ is a basic null sequence.

So by the Squeeze Theorem:
 * $\dfrac 1 {q_{k + 1} q_k} \to 0$

as $k \to \infty$.

So $\left \langle {C_n}\right \rangle$ is indeed a Cauchy sequence.

Hence the result.