Upper bound of P[X < Y] X and Y are independently distributed discrete random variables. is it possible to find an upper bound for P[X<Y] that is always less than or equals to 1?

If $X$ and $Y$ are independent then
$$P[X<Y] = \sum_{\forall x}\sum_{\forall y:y>x} P[x=X]P[y=Y] \leq \sum_{\forall x}\sum_{\forall y} P[x=X]P[y=Y] =1$$
