I've been working through some problems in analysis to try and get a better grasp on the topic. One problem that I came across was the following:
Choose $a_1,a_2,...,a_k \in \mathbb{Q}$ with $a_i > 0$ for all $i$ and $\prod_{i=1}^k a_i > 1$. Starting from any $x_1 > 0$, define a sequence $\{x_n\}$ by the continued fraction $$ x_n = \frac{1}{a_1+\frac{1}{a_2+\frac{1}{\cdots + \frac{1}{a_k + x_{n-1}}}}}$$ Prove that $\{x_n\}$ converges.
This problem is a multiple part problem, where earlier we were asked to prove that if a function $f: I \to I$ on a closed interval $I$ is Lipschitz with Lipschitz constant $M < 1$, then the sequence defined as $\{x_{n+1}\} = f(x_n)$ converges.
Hence I was thinking that to prove the above statement I could prove that the recurrence relation above is Lipschitz with Lipschitz constant $M < 1$, whence convergence follows, but I'm not sure how to algebraically prove it's Lipschitz. Is this a viable solution? How would one prove this?