Let $(X,\mathcal T)$ be a topological space and $(x_d)_{d\in (D,\le)}$ be a net in it and let $a\in X$. Every subnet of $(x_d)_{d\in D}$ has a subnet which converges to $a$. Does $(x_d)_{d\in D}$ converge to $a$?
In fact, $a$ is a cluster point of every subnet as defined in this wikipedia page.
Yes. Proof by contradiction: Suppose $(x_d)_{d\in D}$ didn't converge to $a$. So there is a neighborhood $N$ of $a$ such that this net is frequently outside $N$. That is, for every $d\in D$ there is some $e\in D$ such that $d\leq e$ and $x_e\notin N$. Let $E=\{e\in D:x_e\notin N\}$. Then $(x_e)_{e\in E}$ is a subnet of $(x_d)_{d\in D}$ (because the inclusion map $E\to D$ is cofinal). By assumption, it has a further subnet that converges to $a$, but this is absurd as all its elements $x_e$ lie outside the neighborhood $N$ of $a$.
