  We verify an explicit inequality conjectured recently by Gilmer, thus proving
that for any nonempty union-closed family $F \subseteq 2^{[n]}$, some $i\in
[n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} \approx 0.38$ fraction of
the sets in $F$. One case, an explicit one-variable inequality, is checked by
computer calculation.
