Question. I'm working on proving the Schroder Bernstein THeorem in steps. The first step has me showing that $A_1 \supset C_1 \supset A_2 \supset C_2 \supset \dots$ where $C \subset A$ and there is an injection $f: A \to C$. We also defined $A_1 = A$, $C_1 = C$, and for $n >1$ recursively defined $A_n = f(A_{n-1})$ and $C_n = f(C_{n-1})$.
So with ALL of that in place my simple question came from me doing the proof. I'm doing it via induction, but the only thing that is bothering me is why is $f(A_1) = A_2 \subset C$ explicitly and not $f(A_1) = A_2 \subseteq C$?
