Understanding of the definition of subnets in Folland's Real Analysis

Question

The following is the definition of subnets given in Folland's Real Analysis (page 126)

A subnet of a net $\langle x_{\alpha}\rangle_{\alpha\in A}$ is a net $\langle y_{\beta}\rangle_{\beta\in B}$ together with a map $\beta\mapsto \alpha_\beta$ from $B$ to $A$ such that

• for every $\alpha_0\in A$ there exists $\beta_0\in B$ such that $\alpha_\beta\gtrsim\alpha_0$ whenever $\beta\gtrsim \beta_0$;
• $y_\beta=x_{\alpha_\beta}$.

I have a hard time developing an intuition regarding the first condition.

How is it useful? What is the main use of it?

Would anyone come with a simple example of a net $\langle y_\beta\rangle_{\beta\in B}$ such that the first condition is not satisfied?

Answer 1

It is nothing but the definition of a subsequence .

Note that if $(x_k)$ is a sequence and $(x_{n_k})$ is a subsequence of $(x_k)$ then $n_k$ is an increasing function such that $n_k\ge k$.

Here also if we choose $\alpha_0$ then I can choose some $\beta_0$ satisfying the given property since the above holds.

Answer 2

The condition says that we move further along $A$, the subnet will go there too. It is needed to see that for a convergent net, every subnet converges to that same limit, which is a property known from sequences that we like to keep.

If we leave it out, we could take a constant map to any fixed $\alpha_0$ and get a total trivialisation of the subnet which would converge to that $x_{\alpha_0}$.