The relevant constraint is that cannot hold under these conditions.

proposition:If S is uncountable. Show it is impossible that P({i}) > 0 for every i∈S.

Equivalent statement:Min (P{i}) > 0

Denote this minimum = x

Now note that

So its sufficient to show that

This is true, since you are adding up a strictly positive number an uncountably large number of times (ie, more than 1/x times).