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 $a_n$

take $a_{2n}=1/2n$ and $a_{2n+1}=2n+1$
