Here is an exercise from Resnick - Probability Path (3.18): If $X$ and $Y$ are random variables on $(\Omega, \mathscr{B})$, show that
$$\sup_{A\in\mathscr{B}} |P[X\in A]-P[Y\in A]|\leq P[X\neq Y]$$
Now, I took the definition given by Resnick for $[X\in A]=${$\omega:X(\omega)\in A$}. Moreover, it looks like the set
$$\{\omega: X(\omega)\in A, Y(\omega)\notin A\} \subset \{\omega: X(\omega)\neq Y(\omega)\}$$
Therefore I was thinking to apply in any way the Fatou's lemma, but I really don't know how (perhaps I may use a sequence $A_i$ in order to apply $\sigma$-additivity, but I think it would be too restrictive).
As always, any help would be very appreciated, thanks.
$$[X\in A] \subset [Y\in A] \cup [X\neq Y]$$ so $$P[X\in A] \leq P[Y\in A] +P[X\neq Y].$$ Hence $$P[X\in A]-P[Y\in A] \leq P[X\neq Y] .$$ Similarly $$P[Y\in A]-P[X\in A] \leq P[X \leq Y].$$ Putting these two together we get $$|P[X\in A]-P[X\in A]| \leq P[X \neq Y] .$$
Now take supremum over $A$.