I have an elementary question on nets because I'm not familiar with this concept. Here are two basic facts:
- Every subsequence of a sequence is a subnet;
- Not every subnet of a sequence is a subsequence.
For the second fact, I have seen the following example:
Given a sequence $(x_n)=(x_1,x_2,x_3,x_4,...)$, the net $$(x_\alpha)=(x_1,x_2,x_2,x_3,x_3,...,x_{1+[\frac{n}{2}]},...)$$ is a subnet of $(x_n)$ that is not a subsequence of $(x_n)$.
In this example, $(x_\alpha)$ has a subnet which is a subsequence of $(x_n)$, namely, the sequence $(x_n)$. Could someone give me an example where this doesn't happen?
Explicitly: I'd like an example of sequence $(x_n)$ and a subnet $(x_\alpha)$ of $(x_n)$ such that no subnet of $(x_\alpha)$ is a subsequence of $(x_n)$.
Motivation for the question: I have a bounded sequence in the dual of a normed space. If the space was separable, then I could pass to a weak-* convergent subsequence. However the space is not separable. So, all I have is a subnet weak-* convergent. Presumably, I can't pass to a subsequence. As I said, I'm not familiar with the concept of net and thus I'd like to see an example where the existence of the subsequence fails.
This question and answer give a concrete example of a sequence $(\delta_n)$ in a compact space that has no convergent subsequence (so this compact space is not sequentially compact). The answer gives a "concrete" (if you believe ultrafilters are concrete) subnet $(x_d), d \in D$ that converges to some $f_\mathcal{U}$. This subnet cannot have a subsequence (that is also a subnet!) that converges, because this would be a subsequence of the original sequence that would converge (and this cannot be). It is possible to find $d_n$ that are increasing in the index set $D$ of the subnet, but this is not a subnet of the subnet (as they will not be cofinal).
A subsequence of a sequence is a subnet as well, but a subnet need not have a cofinal subsequence.