Suppose for one $B\subset A$, there is an injection $f:A\to B$. Inductively define a sequence $(C_n)$ of subsets of $A$ by $C_0=A\setminus B$
and $C_{n+1}=f(C_n)$.
Now let $C=\bigcup_{k=0}^\infty C_k$, and define $h:A\rightarrow B$ by
$$h(z)=\begin{cases} f(z), & z\in C \\ z, & z\notin C \end{cases}$$ Prove that $h$ is injective.