  We prove the weight elimination direction of the Serre weight conjectures as
formulated by Herzig for forms of $U(n)$ which are compact at infinity and
split at places dividing $p$ in generic situations. That is, we show that all
modular weights for a mod $p$ Galois representation are contained in the set
predicted by Herzig. Under some additional hypotheses, we also show modularity
of all the "obvious" weights.
