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Homework Help: How to prove the following defined metric space is separable

  1. Oct 1, 2015 #1
    • Member warned about posting without the homework template
    Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to find a countable dense set in ##\mathbb{X}##.
     
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  3. Oct 1, 2015 #2

    Krylov

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    Depending on whether or not this is an exercise question, you may be able to give a quick answer by noting that the dual space of ##X## is ##\ell_1##.
     
  4. Oct 1, 2015 #3

    Dick

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    Yes, that's exactly what you should do. Any thoughts? To get started do you know a countable dense subset of ##\mathbb{R}##?
     
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