Simple Infinite Continued Fraction Converges

Theorem
Let $C = (a_0, a_1, \ldots)$ be a simple infinite continued fraction in $\R$.

Then $C$ converges.

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

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

Let $\epsilon > 0$.

For $m > n \ge \max \set {5, \dfrac 1 \epsilon}$:

So $\sequence {C_n}$ is indeed a Cauchy sequence.

Hence the result.