Showing that the set of convergent sequences is not separable

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SUMMARY

The metric space of convergent complex sequences under the sup norm is not separable. This conclusion arises from the observation that while individual sets of complex sequences converging to specific numbers are separable, their uncountable union results in a non-separable space. The discussion emphasizes the complexity of proving non-separability compared to separability, highlighting the need for effective strategies in such proofs.

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  • Understanding of metric spaces and their properties
  • Familiarity with the sup norm in functional analysis
  • Knowledge of convergent sequences in complex analysis
  • Experience with concepts of separability in topology
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  • Research the properties of metric spaces and their separability
  • Study the sup norm and its implications in functional analysis
  • Explore examples of separable and non-separable spaces in topology
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antm88
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Hi,
I'm looking to show that the metric space of convergent complex sequences under the sup norm is not separable; that is what I assume it is since I cannot find a way to prove that it is separable (I am unable to find any dense subsets).

A set of complex sequences convergent to a certain number is separable, but this case would be an uncountably infinite union of such sets, which makes it seem as if it would not be separable.

It seems much harder in general to prove a space is not separable than to prove that one is separable. Does anyone have any general tips on how to go about proving this sort of thing?

Thanks for any help,
Anthony
 
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