$\begingroup$

Suppose we are given three finite, non-empty sets $A, B, C$ such that $|C|\leq |A| \leq |B|$. Is it always true that $\frac{|A\cap C|}{|A|} + \frac{|B\cap C|}{|B|} \leq 1 + \frac{|A\cap B|}{|B|}$?

I have tried countless examples for days and I haven't found any counterexample so far... This is just a curious question. But if it's true, there has to be a proof, right?