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!
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What is a simple set? – M. Winter Jul 24 '18 at 14:03
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What is $A$ restricted to $2e?$ Please use MathJax to format questions on this site. – saulspatz Jul 24 '18 at 14:19
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Think about what happens as $e$ goes to infinity (or as $i$ goes to infinity) ... – Noah Schweber Jul 25 '18 at 01:48
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For every $e\in\Bbb N$, a subset of $A^\complement$ has cardinality $>e$, so $|A^\complement|>e$.
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