Let $m$ be the integer so that $\left(\frac{m}{2}\right) < n \le \left(\frac{m+1}{2}\right)$. Then since $m$ lines intersect in at most $\left(\frac{m}{2}\right)$ points, we have $n > m$. We shall show that there exist $m + 1$ lines that intersect in exactly $n$ points. Let $p = n - \left(\frac{m}{2}\right)$ and $q = \lfloor (m-p)/2 \rfloor$.

Case (i): $m-p$ is even. Draw $q$ pairs of lines so that they intersect in $q$ distinct points on the $x$-axis. Draw another $p$ lines, not parallel to the $x$-axis, so that the $2q+p=m$ lines are pairwise nonparallel. Then draw another line on the $x$-axis. These $m+1$ lines intersect in exactly $n$ points.

Case (ii): $m-p$ is odd. Draw $q$ pairs of lines so that they intersect in $q$ distinct points on the $x$-axis. Draw another $p+1$ lines so that exactly one of them is parallel but not on the $x$-axis, and that the $2q+p+1=m$ lines are pairwise nonparallel. Then draw another line on the $x$-axis. These $m+1$ lines intersect in exactly $n$ points.