# Showing that the set of convergent sequences is not separable

1. Dec 30, 2009

### antm88

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