Let $A=\{0,1,\dots\}$ and $B=\{n: P(X=n)\ge P(Y=n)\}$. Show that $P(X \in A)-P(Y\in A) \le P(X∈(A∩B)) - P(Y ∈(A∩B))$

Question

$X$ and $Y$ are non-negative integer valued RV's. I don't know how to even approach this question! I was thinking of approaching like this:

$P(X\in A) = P(X\in A\mid X\in(A ∩ B))\cdot P(X \in (A ∩ B)) \tag{1}$ and $P(Y\in A) = P(Y∈A \mid Y∈(A ∩ B))\cdot P(Y∈(A ∩ B))\tag{2}$

Then, $(1) - (2) \dots$. But then I got stuck since I don't know what to do!

Please help me to solve this question fully! I need guidance and suggestions. Thanks in advance!