A filter and its canonical derived net have the same limits and cluster points

If a neighbourhood $N$ of $p$ belongs to $\mathcal F$, choose $x\in N$; then $(N,x)$ is an index, and every later $(B,y)$ has $B\subseteq N$, hence $y\in N$. Thus filter convergence implies convergence of the derived net.

If the derived net is eventually in $N$, take a threshold $(A,x)$. Applying eventuality to indices $(A,y)$ with $y\in A$ gives $A\subseteq N$; upward closure of the filter gives $N\in\mathcal F$. Thus convergence is equivalent.

The derived net is frequently in $N$ exactly when every $A\in\mathcal F$ meets $N$: after $(A,x)$ a point of $A\cap N$ supplies a later index, and conversely frequent membership after $(A,x)$ supplies such a point. Therefore its cluster points are exactly those of $\mathcal F$.