Let $a\_{n}$ be a sequence such that $\lim\_{n\to \infty} a\_{n} a\_{n+1}=1$,

Is the following statement true or false:
If $0$ is a partial limit of $a\_{n}$ (for some subsequence $a\_{n\_{k}}$), so does $\infty$.

I think it's true, for example: $a\_{n}=\lbrace \frac{1}{n}, odd| n, even\rbrace$
But haven't found an elegant way to prove it.