My question is in the lemma: If $f: A \rightarrow B$ is injective, where $B \subset A$, then there is a bijection between $A$ and $B$. The author commented that, with $Y = A-B$, it can be shown that $f^k(Y) \cap f^m(Y) = \emptyset$ whenever $0 \le k \lt m$. I understand this part, but I do not see how this step is necessary in proving the lemma, or Bernstein-Schroeder.
- Since $X = \bigcup_{n \ge 0} f^n(Y) \subset A$, $f$ is injective on $X$. Then, it is bijective between $X$ and $f(X)$.
- $A-X = B-f(X)$, which is disjoint from $f(X)$. So, the continuous application of $f$ on $X$ will not "spill into" $A-X$.
Hungerford proved Bernstein-Schroeder differently, but he used partitions. That's why I suspect that the assertion about the disjoint sets in $\{ f^k(Y) \}$ is necessary.