if $\lim_{n \to \infty}a_n = \infty$ then $\lim_{n \to \infty}(-a_n)=-\infty$ if $\lim_{n \to \infty}a_n = \infty$ then $\lim_{n \to \infty}(-a_n)=-\infty$
I know I need to multiply by -1 to get $-a_n$ but I don't know how to formally write this proof. 
 A: A limit $\lim_{n\to\infty}a_n=\infty$ means that for any (arbitrarily large) $L\in\Bbb R$ there is an $n$ so that $a_n>L$, i.e. $a_n$ exceeds any bound. 
Now if you choose a very small (i.e. very negative) number $S\in\Bbb R$, then $-S$ is very big and we can find an $n$ so that $a_n>-S$. We can multiply this by $-1$ and find $-a_n<S$. And this is exactly what $\lim_{n\to\infty}-a_n=-\infty$ means: for any $S\in\Bbb R$ you can find an $n$ so that $-a_n<S$. The sequence $-a_n$ goes below any bound.

Here I will try to give a proof with more words. Maybe this makes more clear what is going on. 
When a sequence diverges to $\infty$, this has a concrete mathematical meaning. It means that it grows and grows and does not stay below any bound. Imagine a number. The sequence will finally exceed this number. Now imagine this sequence as a graph. You see a curve rising. Now flip the curve at the $x$-axis. This gives $-a_n$. Now the sequence is below zero and gets smaller and smaller. Imagine any (negative) number. The sequence will finally go below this number. And this is exactly what is meant when saying that it diverges to $-\infty$.
A: Assume $\displaystyle \lim_{n \to \infty} a_n = \infty$, i.e. $\forall L \exists N \forall n>N:a_n>L$.
We now have to prove $\displaystyle \lim_{n \to \infty} (-a_n) = -\infty$, i.e. $\forall L \exists N \forall n>N:-a_n<L$.
This is trivial.
