I'm trying to show that the continued fraction $[1,1,\dots]$ converges. After that it is easy to determine the limit, so I'm interested in a proof of convergence specifically.
I don't think it's difficult to prove using Banach's fixed point theorem, but I'm looking for a more elementary proof, hopefully one that doesn't use more than basic results about infinite sequences.
Continued fractions converge by their structural properties. If we have $$ \alpha=[a_0;a_1,a_2,a_3,\ldots]$$ with $a_0\in\mathbb{N}$ and $a_{>0}\in\mathbb{N}^+$, by denoting through $$ \frac{p_n}{q_n}=[a_0;a_1,a_2,\ldots,a_n] $$ the $n$-th convergent of $\alpha$ we have that $\frac{p_{n+2}}{q_{n+2}}$ always belongs to the interval whose endpoints are $\frac{p_n}{q_n}$ and $\frac{p_{n+1}}{q_{n+1}}$. Additionally $\left|\frac{p_n}{q_n}-\frac{p_{n+1}}{q_{n+1}}\right|$ exactly equals $\frac{1}{q_n q_{n+1}}$ and $\{q_n\}_{n\geq 1}$ is a rapidly (exponentially) increasing sequence. It follows that both $\left\{\frac{p_{2n}}{q_{2n}}\right\}_{n\geq 1}$ and $\left\{\frac{p_{2n+1}}{q_{2n+1}}\right\}_{n\geq 0}$ are monotonic, convergent, and convergent to the same limit, $\alpha$.