# Subnet with an infinite directed set

I have a question about nets.

I consider an infinite directed set $$(A, \prec_A)$$ and a net $$\{x_a\}_{a \in A}$$ in a compact topological space $$X$$. Then, it admits a convergent subnet (I consider Kelley subnets here). My question is the following: is there always exists one of these subnets $$\{y_b\}_{b \in B}$$ such that the directed set $$(B, \prec_B)$$ is infinite ?

• All subnets of $A$ will be infinite, except for trivial cases... – Henno Brandsma Mar 20 '20 at 22:56
For $$y: (B, \prec_B) \to X$$ to be a subnet of $$x: (A, \prec_A) \to X$$, when $$A$$ is infinite, $$B$$ has to be infinite too, iff $$A$$ satisfies the requirement that all tails $$T(a_0)=\{a: a_0 \prec a\}$$ are infinite: If not, such a finite tail defines a finite subnet itself, and otherwise $$B$$ has a cofinal image in $$A$$ (under the connecting map) and in a directed set a finite set can only be cofinal if it contains a maximum of the set, while we assumed all tails to be infinite.