absolute value of supremum is smaller than or equal to supremum of absolute values Is the following statement correct? 
"The absolute value of supremum of a set is smaller than or equal to the supremum of the absolute values of the first set"
It would be helpful to proof that the absolute value of the integral of a function is smaller than or equal the integral of the absolute value.

Yes. Because $a\leq |a|$, $\sup A\leq \sup|A|$, so if $\sup(A)\geq 0$ you would be done.  If $\sup(A)<0$, then $|A|=-A$ and $|\sup A|=-\sup(A)=\inf(-A)=\inf |A|\leq \sup|A|$.
Here $|A|=\{|a|:a\in A\}$, and $-A=\{-a:a\in A\}$.
