Is the difference of two decreasing functions also decreasing? If I have two decreasing functions $f(x)$ and $g(x)$, is their difference
$ H(x)=f(x)-g(x) $ also a decreasing function?

Not necessarily, especially if by decreasing you mean "$x > y$ implies $f(x) < f(y)$".  
Take $f(x) = -x$ and $g(x) = -x$.  Both are decreasing in the sense I've described above, but $H(x) = 0$ for all $x$.  Then $H(x)$ is not decreasing, since for all $x, y$ with $x > y$, $H(x) = H(y) = 0$.
