  In 1984 Brown and Geoghegan proved that Thompson's group $F$ is of type
$\textrm{F}_\infty$, making it the first example of an infinite dimensional
torsion-free group of type $\textrm{F}_\infty$. Over the decades a different,
shorter proof has emerged, which is more streamlined and generalizable to other
groups. It is difficult, however, to isolate this proof in the literature just
for $F$ itself, with no complicated generalizations considered and no
additional properties proved. The goal of this expository note then is to
present the "modern" proof that $F$ is of type $\textrm{F}_\infty$, and nothing
else.
