Prove max f(x) = -min -f(x) How do I prove : 
$$max f(x)= - min -f(x)$$
I am trying to prove this, and have tried to use my book but I am stuck. 

Let $A\subset\mathbb R$ have a maximum; i.e. some
$m\in A$ exists such that $a\leq m$ for each $a\in A$.
If $-A$ denotes the set $\left\{ -a\mid a\in A\right\} $ then $-m\in-A$
and $-m\leq-a$ is true for each $a\in A$. 
This tells us that $-m$ is a minimum
of $-A$.
Denoting $m=\max A$ and $-m=\min\left(-A\right)$ we have: $$\max A=m=-\left(-m\right)=-\min\left(-A\right)$$
