Proving that $\liminf (1/a_n) = 1/ \lim \sup (a_n)$ for any positive sequence $(a_n)$?
Let $A \subseteq (0,\infty$) such that $\inf A>0$. Let $\frac{1}{A}=\{\frac{1}{a},a\in A\}$. Prove that $\sup \frac{1}{A}=\frac{1}{\inf A}$.