Proof that absolute value of supremum equals suprememum of absolute value

Question

Can someone prove that |sup(A)|=sup|A| ? Please and thanks.

Answer 1

If $A=(-\infty,0)$. $|\sup A|=0$, but $\sup|A|=\infty$.

Answer 2

It is false: if $A=[-1,0]$, then $\sup A=0=\lvert\sup A\rvert$, while $\lvert A\rvert=[0,1]$ and $\sup\lvert A\rvert=1$.