Q: Ιnequality relationship Let $a,b,c,d$ positive numbers. They are connected with the relations $$b<d,\quad  a<c,\quad b<a,\quad d<c$$
Is it possible to prove that $a-b<c-d$?

A: No, because the result might not be true. Take, for example, $a=5$, $b=1$, $c=6$, $d=3$. Then $b<d$, $a<c$, $b<a$, and $d<c$, but $a-b=4 \not< 3=c-d.$ 
If instead of $b<d$ you had $b>d$, then the result would be true. Multiplying by $-1$, you get $-b<-d$. Then $a-b<c-b<c-d$.
