Let us say that $x$ and $y$ belong to an interval $I$ . Assume that the function $g$ is bounded on I . is it true that $|g(x)-g(y)|$ is less than or equal to $\sup\{g(x)-g(y)\}$?? Of course this is trivial if the absolute value sign is removed ..Its seems that my Analysis book makes this assumption when proving certain facts concerning integrability but I just do not know if this is true or if it is how to prove it.....