# 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.

• The link I provided as a "duplicate" was a question asked only about the equality of both values, which doesn't hold. I reopened the question because you have posed a question asking about an inequality (namely, whether $$\vert\sup A\vert \leq \sup \{|a|\mid\;a \in A\}$$ Mar 23, 2017 at 18:15
• I saw and analyzed this question before posting this one, but any check is welcome so thank you :) Mar 23, 2017 at 20:25

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 $-A=\{-a:a\in A\}$.