let $\{ a_n \}$ be a sequence where for each $n \in \mathbb N$ $ a_n \neq 0 $ and where $\lim_{n \rightarrow \infty} a_n a_{n+1} = L$ with $L \neq 0$
I want to prove that
$\lim_{n \rightarrow \infty} a_n a_{n+3} = L$
and that
$\lim_{n \rightarrow \infty} a_n a_{n+2} \neq -1$
Any ideas?
Thanks!
Edit: Intuitively it's clear but I am looking for a real regorous proof..