I have the following sum:
$$F=\sum_{i=1}^N p_i g_i .$$
Where $g_i$ is random variable and $p_i$ is function of $g_i$ such that, if $g_i<0.01 , p_i=0$.
On the other hand, I have (CONSTRAINT):
$$\sum_{i=1}^N p_i =NP$$
So if we have more $g_i<0.01$ this leads to more $p_i=0$ but the non-zero $p_i$ become greater.
I want to prove that with each further $p_i=0$, F will decrease even if the Constraint is still satisfied.