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May 17, 2022 at 7:15 comment added Jochen Wengenroth The sequence $x_{2n}=n$ and $x_{2n+1}=0$ satisfies $\bigcap_{n\in\mathbb N}\overline{\{x_k:k\ge n\}}=\{0\}$ but it diverges. I don't know whether this simple (and maybe superficial) functorial interpretation of convergence appears somewhere in the literature.
May 16, 2022 at 17:29 comment added Davi Barreira Why does it not yet characterize convergence of sequences? Would you know if this has been worked out somewhere?
Sep 30, 2020 at 11:11 history answered Jochen Wengenroth CC BY-SA 4.0