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?

The least upper bound is $1$.
First, probabilities do not exceed $1$, so this is a trivial upper bound. However, we can achieve such a value, so it is the least upper bound.
Let $X$ be uniform on $\{1,2\}$, and let $Y$ be uniform on $\{3,4\}$.
(So $P(X=1)=P(X=2)=P(Y=3)=P(Y=4)=0.5$.)
Then $P(X<Y)=1$.
