Showing that the set of convergent sequences is not separable

In summary, the speakers are discussing the separability of the metric space of convergent complex sequences under the sup norm. They are unsure if this space is separable and are seeking tips on how to prove its separability. One speaker mentions that proving a space is not separable is more difficult than proving it is separable.
  • #1
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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Related to Showing that the set of convergent sequences is not separable

What is a convergent sequence?

A convergent sequence is a sequence of numbers that approaches a specific limit as the number of terms in the sequence increases. The limit is the value that the sequence approaches and is denoted by the symbol "L".

What does it mean for a set of convergent sequences to be separable?

A set of convergent sequences is separable if it can be divided into two non-overlapping subsets, such that every sequence in the set belongs to one of the subsets. In other words, there is a clear distinction between the sequences in the set.

Why is it important to show that the set of convergent sequences is not separable?

Showing that the set of convergent sequences is not separable is important because it helps us understand the structure and properties of the set. It also has implications in other areas of mathematics, such as analysis and topology.

How can you prove that the set of convergent sequences is not separable?

To prove that the set of convergent sequences is not separable, you can use a proof by contradiction. Assume that the set is separable and then show that this assumption leads to a contradiction. This would prove that the set is not separable.

What are some examples of sets of convergent sequences that are not separable?

One example is the set of all sequences of rational numbers that converge to an irrational number. Another example is the set of all sequences of real numbers that converge to a non-rational number.

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