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Here on page 232-233 the author offers a proof of the Bernstein-Schroeder Theorem. He uses the subset

$$\bigcup_{k=0}^\infty(g\circ f)^k(A-g(B))\subseteq A$$

and I'm not exactly sure how to parse this. Does this mean the union of the powers of the composition $g\circ f$? If so how is this a subset of $A$? I'm having a bit of trouble seeing it.

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It seems like you have the right interpretation. Notice that $g\circ f$ is a function sending $A$ to $A$, and therefore so is $(g\circ f)^k$ for all $k$.

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