Show that $\limsup(a_n)\geq1$ if $a_n$ is bounded, positive, and $\lim_{n\to\infty} a_na_{n+1}=1$

Question

Let $\{a_n\}$ be a bounded positive sequence ($a_n>0$).

$$\lim_{n\to\infty} a_na_{n+1}=1.$$

Show that $\limsup(a_n)\geq1$.

Thanks.