$\displaystyle \lim\sup x_n=-\infty\implies \lim x_n=-\infty,$ does the converse hold?

How to prove this?

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- Mar 29th 2011, 03:33 PMConnectedDivergence of limit superior
$\displaystyle \lim\sup x_n=-\infty\implies \lim x_n=-\infty,$ does the converse hold?

How to prove this? - Mar 29th 2011, 05:43 PMDrexel28
- Mar 29th 2011, 05:55 PMBeaky
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 $\displaystyle \lim_{n\to\infty}x_{n} = -\infty$