  The empirical upper limit to Red Supergiant (RSG) luminosity, known as the
Humphreys-Davidson (HD) limit, has been commonly explained as being caused by
the stripping of stellar envelopes by metallicity-dependent, line-driven winds.
As such, the theoretical expectation is that the HD limit should be higher at
lower metallicity, where weaker mass-loss rates mean that higher initial masses
are required for an envelope to be stripped. In this paper, we test this
prediction by measuring the luminosity function of RSGs in M31 and comparing to
those in the LMC and SMC. We find that $\log (L_{\rm max}/L_{\odot}) = 5.53 \pm
0.03$ in M31 (Z $\gtrsim$ Z$_{\odot}$), consistent with the limit found for
both the LMC (Z $\sim$ 0.5 Z$_{\odot}$) and SMC (Z $\sim$ 0.25 Z$_{\odot}$),
while the RSG luminosity distributions in these 3 galaxies are consistent to
within 1$\sigma$. We therefore find no evidence for a metallicity dependence on
both the HD limit and the RSG luminosity function, and conclude that
line-driven winds on the main sequence are not the cause of the HD limit.
