A simple argument is presented by which one can show that the critical inverse temperature beta-c of a two-dimensional Coulomb gas (standard or hard-core) with activity z satisfies beta-c-less-than-or-equal-to-beta, where beta = beta-(z) --> (1 + square-root-3)8-pi/(3-square-root-3) in the low-activity limit. Previous results yield beta-(z) --> 24-pi.