Let $A,B$ be subset of $S^n$, $n\geq 2$. Show that if $A$ and $B$ are closed, disjoint, and neither separates $S^n$, then $A\cup B$ does not separate $S^n$.
I've thought to do it by contradiction and use the Mayer-Vietoris exact sequences, but really I have some doubts, so the appreciate any help, thanks.