If $(\limsup_{n\to\infty} x_{n})\cdot(\limsup_{n\to\infty} 1/x_{n})=1$, then the sequence $x_n$ converges
Proving $\limsup\frac 1 {a_n}=\frac 1 {\liminf a_n}$ and $\limsup a_n\cdot \limsup \frac 1 {a_n} \ge 1$