How to show that $\lim_{n\to\infty}[a_1,\cdots,a_n]$ exists if $a_k\geq 2$ for all $k$? 
Consider a sequence of positive real numbers $(a_n)$. Define $[a_1]=\frac{1}{a_1}$ and recursively inductively $[a_1,\cdots,a_n]=\frac{1}{a_1+[a_2,\cdots,a_n]}$. Suppose $a_k\geq 2$ for all $k$. How to show that  $\lim_{n\to\infty}[a_1,\cdots,a_n]$ exists?

I was trying to show that the sequence is monotone, which is not true. A special related case is done here, which is not very helpful to have a generalization. 

[Added to answer the confusion in comments.] The definition above should be understood properly as follows. For any positive real number $a$, define $[a]:=\frac{1}{a}$. Now, given any two positive real numbers $a_1,a_2$, one can define $[a_1,a_2]:=\frac{1}{1+[a_2]}$. One can thus keep going on in this fashion to define $[a_1,a_2,\cdots,a_n]$. To write down a few terms explicitly, 
$$
[a_1,a_2]=\frac{1}{a_1+\frac{1}{a_2}},\ 
[a_1,a_2,a_3]=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3}}},\
[a_1,a_2,a_3,a_4]=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{a_4}}}},\cdots
$$
 A: You only need $a_i \ge 1$.
Define
$$\begin{bmatrix} p_{-1} & p_{0} \\ q_{-1} & q_{0} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. $$
Recursively, set
$$p_k = a_kp_{k-1} + p_{k - 2}, q_k = a_kq_{k-1} + q_{k - 2} \tag{1} $$
for $k \ge 1$. We will show that
$$ [a_1,\dots,a_n,a_{n+1}] = \frac{a_{n+1}p_n + p_{n - 1}}{a_{n+1}q_n + q_{n - 1}}. $$
This holds by induction:
\begin{align}
[a_1,\dots,a_n,a_{n+1}] &= \left[ a_1,\dots,a_n + \frac{1}{a_{n+1}} \right] \\
&= \frac{\left(a_n + \frac{1}{a_{n+1}} \right)p_{n-1}+p_{n-2}}{\left(a_n + \frac{1}{a_{n+1}} \right)p_{n-1}+p_{n-2}} \\
&= \frac{(a_np_{n-1}+p_{n-2)} + \frac{1}{a_{n+1}}p_{n - 1}}{(a_nq_{n-1}+q_{n-2}) + \frac{1}{a_{n+1}}q_{n - 1}} \\
&= \frac{a_{n+1}p_n + p_{n - 1}}{a_{n+1}q_n + q_{n - 1}}
\end{align}
and the base case is easy to verify.
Thus we get the matrix equation
$$ \begin{bmatrix} p_n & p_{n-1} \\ q_n & q_{n - 1} \end{bmatrix} \begin{bmatrix} a_{n+1} & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} p_{n+1} & p_{n} \\ q_{n+1} & q_{n} \end{bmatrix} $$
and by induction
$$ \begin{bmatrix} p_{n+1} & p_{n} \\ q_{n+1} & q_{n} \end{bmatrix} = \begin{bmatrix} a_{n+1} & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} a_{n} & 1 \\ 1 & 0 \end{bmatrix} \cdots \begin{bmatrix} a_{1} & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $$
taking determinants we have
$$ p_{n+1}q_n - p_nq_{n+1} = (-1)^{n}. $$
Hence
$$ \left\lvert \frac{p_{n+1}}{q_{n + 1}} - \frac{p_n}{q_n} \right\rvert = \frac{|p_{n+1}q_n - p_nq_{n+1}|}{q_{n+1}q_n} = \frac{1}{q_{n+1}q_n}. $$
From $(1)$ and the assumption that $a_k \ge 1$ we have $q_n \ge n$. Thus the sequence $p_n/q_n$ is Cauchy and converges to some limit.
Reference
Automatic Sequences by J-P. Allouche and J. Shallit, Cambridge University Press (2003), pages 44-46.
A: In Khinchin's book "Continued Fractions" it is shown that
$$
\lim_{n\to\infty} [a_0;,a_1, \ldots, a_n]
$$
exists if and only if $\sum_{n=1}^{\infty} a_n = \infty$, where $(a_n)_n$ is a sequence of positive numbers (Theorem 10). This solves your problem, since your sequence of positive numbers has a uniform lower positive bound.
