Divergence of limit superior

• March 29th 2011, 03:33 PM
Connected
Divergence of limit superior
$\lim\sup x_n=-\infty\implies \lim x_n=-\infty,$ does the converse hold?

How to prove this?
• March 29th 2011, 05:43 PM
Drexel28
Quote:

Originally Posted by Connected
$\lim\sup x_n=-\infty\implies \lim x_n=-\infty,$ does the converse hold?

How to prove this?

You tell me, I've helped you with a bunch. Consider maybe the fact that $\lim x_n\leqslant \limsup x_n$ assuming that $\lim x_n$ converges to some element of $\mathbb{R}\cup\{-\infty,\infty}$
• March 29th 2011, 05:55 PM
Beaky
If you understand the definition then this is basically trivial.

$\limsup x_{n} = \lim_{n\to\infty} \sup \{x_{i}\ : i \geq n\}$

Then use the definition of $\lim_{n\to\infty}x_{n} = -\infty$