Is this property regarding infinite sets true?
As bof points out, if you mean $S-A=\{s-a:s\in S, a\in A\}$, then it's true that $\bigcup_{n=1}^\infty(S-A_n)\subset \bigcup_{n=1}(S-A_n)$. Since $a \in \bigcup_{n=1}^\infty(S-A_n) \implies a\in S - A_n\text{(for some n)}\implies a=s-a_n\in \bigcup_{n=1}(S-A_n)$. The same for the converse.