Is it possible that $L=0$ is a partial limit of $a_n$? Given: $$\lim_{n\to\infty}(a_na_{n+1})=1$$  
Is it possible to say that $L=0$ is a partial limit of $a_n$?
I don't see how we can show such a thing because the limit of $a_{n+1}$ will always be the limit of $a_n$

What about the sequence
$$2,\frac{1}{2}, 3, \frac{1}{3}, \dots, n, \frac{1}{n}, n+1, \frac{1}{n+1}, \dots $$
