Something confusing in the proof of existence of simple sets in Soarse's book.

Question

In Soares's book he tries to prove that there's a simple set and I understand it well but there's something confuses me when he proves that the complement of $A$ is infinite he says that $\mathrm{card}(\text{complement of $A$ restricted to $2e$})\ge e+1$ therefore the complement of $A$ is infinite!!!!! Or $A$ intersection $[0, 2i) \le i$ then the complement of $A$ is infinite!! I cant figure out why it's infinite??? It's so confusing to me. Please help if you can. Thanks!

Answer 1

For every $e\in\Bbb N$, a subset of $A^\complement$ has cardinality $>e$, so $|A^\complement|>e$.