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 Apr14 awarded Supporter Mar29 awarded Nice Question Sep24 awarded Autobiographer Jun23 accepted Path-connected and locally connected space that is not locally path-connected Jun23 comment Path-connected and locally connected space that is not locally path-connected Ingenious! Except there's one little problem with your proof of the lemma. The set $U_b = b \cup (A \setminus B)$ may not be open, since it's possible that $B$ is not countable. So we should choose a countable subset $B_0$ of $B$ which has more than 1 element, and let $U_b = b \cup (A \setminus B_0)$. Jun23 comment Path-connected and locally connected space that is not locally path-connected Thanks. But that site doesn't have an example having these properties. Jun22 asked Path-connected and locally connected space that is not locally path-connected Oct7 revised Definition of $\pi$, $\lim_{n \to \infty}{n \sin(180^o/n)}$ add tag calculus Oct7 asked Definition of $\pi$, $\lim_{n \to \infty}{n \sin(180^o/n)}$ Oct4 comment Please prove: $\lim_{n\to \infty}\sqrt[n]{\frac{1}{n!}} = 0$ @mick: I'm so sorry to ask so vaguely, but Alfred just happened to show what I want to know. It is exactly the answer on my book but more detailed. And now I can understand the book's answer. Oct4 comment Please prove: $\lim_{n\to \infty}\sqrt[n]{\frac{1}{n!}} = 0$ @robjohn: Thank you for telling me this. I'll post more next time I ask. Oct3 accepted Please prove: $\lim_{n\to \infty}\sqrt[n]{\frac{1}{n!}} = 0$ Oct3 asked Please prove: $\lim_{n\to \infty}\sqrt[n]{\frac{1}{n!}} = 0$ Sep16 awarded Student Sep16 awarded Scholar Sep16 accepted A Dedekind infinite set has a countably infinite proper subset Sep16 awarded Editor Sep16 revised A Dedekind infinite set has a countably infinite proper subset added 9 characters in body Sep16 asked A Dedekind infinite set has a countably infinite proper subset