A point is a cluster point of a net if and only if some subnet converges to it

Suppose $p$ is a cluster point. Let $E=\{(d,N):N\in\mathcal N(p),\ d\in D,\ x_d\in N\}$, ordered by $(d,N)\preceq(e,M)$ when $d\le e$ and $M\subseteq N$.

Conversely, suppose a subnet $y_e=x_{\phi(e)}$ converges to $p$. Given a neighbourhood $N$ and $d\in D$, choose $e_0$ after which $y$ lies in $N$ and choose $e_1$ after which $\phi(e)\ge d$; a common upper bound $e$ of $e_0,e_1$ gives $\phi(e)\ge d$ and $x_{\phi(e)}=y_e\in N$. Hence $x$ is frequently in $N$.

The set $E$ is directed: for $(d,N),(e,M)\in E$, take $h\ge d,e$ in $D$; frequent membership in $N\cap M$ gives $k\ge h$ with $x_k\in N\cap M$, and $(k,N\cap M)$ is above both pairs.

Put $y_{(d,N)}=x_d$ and $\phi(d,N)=d$. For every $d_0\in D$, the pair $(d_0,X)$ lies in $E$, and every later pair has first coordinate at least $d_0$. Thus $\phi$ is eventually cofinal and $y$ is a subnet of $x$.

For a neighbourhood $N$ of $p$, choose $(d,N)\in E$ using frequent membership in $N$. Every pair later than it has second coordinate contained in $N$, hence its $y$-value lies in $N$. Thus $y\to p$.

Steps 1.1 and 2.1--2.3 construct a convergent subnet from a cluster point, and step 1.2 gives the converse.