Understanding the Limit Superior and Limit Inferior of a Sequence

[wrldt]

Let $\{a_n\}$ be a sequence of real numbers.

Then $\liminf\limits_{n \rightarrow \infty}$ $a_n = \limsup\limits_{n \rightarrow \infty}$ $-a_n$

So my first strategy was to translate these into the definitions given in Rudin:

$\limsup\limits_{n \rightarrow \infty}$ $a_n = \sup_{n \geq 1}(\inf_{k \geq n} a_k)$

From a homework problem in the first chapter, we know that $\inf$$A=-\sup$ $-A$

So applying that reasoning, I get:

$\sup_{n \geq 1}(-\sup_{k \geq n} -a_k)$

But now I'm not sure how to mess with that sup in the front.

 

[kurt.physics]

I know that \sup_{m \in A} f(m) = -\inf_{m \in A}(-f(m)).But I'm not sure how to apply that here, since the -f is already inside the -a_k.Any help would be appreciated.