Let be a non-negative sequence of monotone decreasing integrable functions on .
Show that 0 if and only if 0.
Since the functions are non-increasing (monotome decreasing) it means that for each . Therefore, . However, we are told that therefore the integral, being squeezed between those two sequences, must go to zero too. Conversely it is not true. Just let with for . Then each is integrable, non-negative, non-increasing with zero integral however does not converge to the zero function.