A point $c\in X$ is a cluster point of the net $(x_d)_{d\in D}$ if, for every neighborhood $U$ of $c$ and for any $d_0\in D$ there exists $d\ge d_0$ such that $x_d\in U$. In the other words, $x_d$ is frequently (cofinally) in $U$.
Question: How to show that for any cluster point $c$ of $(x_d)_{d\in D}$ there is a subnet converging to $c$?
Since this result is often used in connection with nets, I considered useful to have the proof available somewhere on the site.
It is worth mentioning that different definitions of subnet are commonly used: Different definitions of subnet. (Although for our purpose they are similar in the sense that they give the same set of limits of convergent subnets.)
Let us define the directed set $$D'=\{(U,d); U\text{ is a neighborhood of }c, d\in D, x_d\in U\}$$ with the natural ordering, i.e., $$(U,d)\le (U',d') \Leftrightarrow (U\supseteq U') \land (d\le d').$$ This is indeed a directed set, if $x_{d_1}\in U_1$ and $x_{d_2}\in U_2$ then there is $d\in D$ such that $d\ge d_1$, $d\ge d_2$ and $x_d\in U_1\cap U_2$.
Let us define $f\colon D'\to D$ by $f(U,d)=d$. The map $f$ is clearly cofinal and monotone. We see that $x_{(U,d)}=x_d$ and that $(x_{(U,d)})$ is a subnet of $(x_d)$.
Two or three various notions of subnet are commonly used.1 But we have found a map which is both monotone and cofinal - which is the most restrictive of the three definitions.
1I wrote two or three since AA-subnet is mentioned less frequently, so this definition is probably less common. See, for example: Different definitions of subnet on this site, my notes on various definition of subnets which are based on Schechter's book.