Let $(H_*, \partial_*)$ be a homology theory satisfying the dimension axiom. Let $A \subset S^n$ be a proper subset. Show that $H_n(S^n, A)$ is not trivial.
I tried applying the long exact sequence with no success, since $A$ could be virtually anything. I assume $H_n(A) \to H_n(S^n)$ is not injective either, because then I'd get a short exact sequence, but it can not split.
Anything obvious that I missed?