TITLE: Intersection of infinite sets is infinite?
QUESTION [7 upvotes]: 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?

REPLY [7 votes]: This argument is not true...a Counter example is the set:$$\left\{\frac 1n ,\ n\in \mathbb N\right\}$$