I know that if $C \subseteq [0,1]$ is uncountable, then there exists $a \in (0,1)$ such that $C \cap [a,1]$ is uncountable. Is it still true for any infinite sets? That is, if $C \subseteq [0,1]$ is infinite, does there exist an $a \in (0,1)$ such that $C \cap [a,1]$ is infinite?
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Not necessarily. Consider $C := \{\frac{1}{n} : n \in \mathbb{N}\}$. |
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Not true. Take $$C = \left\{1, \frac12, \frac13, \frac14, \ldots , \frac1N, \ldots \right\}.$$ [Jeez, everyone came up with the same example at once] |
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Every set E such that : $$\forall \epsilon >0 \quad E\cap(\epsilon,1)\text{ is finite}$$ is a counter example. You just need $0$ to be an accumulation point of your set. You can take $$\left\{\frac 1n ,\ n\in \mathbb N\right\}$$ for instance. |
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This argument is not true...a Counter example is the set:$$\left\{\frac 1n ,\ n\in \mathbb N\right\}$$ |
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