  Deriving necessary and sufficient conditions for a scalar potential to be
bounded from below (BFB) is a difficult task beyond the simplest cases.
Recently, a set of BFB conditions was proposed for the $A_4$-invariant
three-Higgs-doublet model (3HDM). However, that set of conditions relied on
numerical scan, and a complete analytic proof was lacking. Here, we fill this
gap. We prove that the conjectured BFB conditions are indeed necessary and
sufficient within the neutral Higgs subspace. We bypass technically challenging
direct algebraic computations with a novel technique that relies on an
auxiliary function, which is related to the Higgs potential but which is easier
to analyze. This technique may finally be sufficient to tackle the more
involved case of the original Weinberg's 3HDM model.
