Consider a sequence of positive real numbers $(a_n)$. Define $[a_1]=\frac{1}{a_1}$ and
recursivelyinductively $[a_1,\cdots,a_n]=\frac{1}{a_1+[a_2,\cdots,a_n]}$. 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, $$ [a_1,a_2]=\frac{1}{a_1+\frac{1}{a_2}},\ [a_1,a_2,a_3]=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3}}},\ [a_1,a_2,a_3,a_4]=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\frac{1}{a_4}}}},\cdots $$