$\displaystyle \lim\sup x_n=-\infty\implies \lim x_n=-\infty,$ does the converse hold?
How to prove this?
If you understand the definition then this is basically trivial.
$\displaystyle \limsup x_{n} = \lim_{n\to\infty} \sup \{x_{i}\ : i \geq n\}$
Then use the definition of $\displaystyle \lim_{n\to\infty}x_{n} = -\infty$