infimum and supremum of sum and product

Suppose that the real functions $f$ and $g$ are defined on an interval $\mathrm{\Delta }$.  Then on this interval

• $inf(f+g)\geqq inff+infg$

• $sup(f+g)\leqq supf+supg$

If $f$ and $g$ are also nonnegative on $\mathrm{\Delta }$, we can write

• $inf(fg)\geqq inff\cdot infg$

• $sup(fg)\leqq supf\cdot supg$