  It is shown that the absolute value of the persistent current in a system
with toroidal geometry is rigorously less than or equal to $e \hbar N /4 \pi m
r_0^2$, where $N$ is the number of electrons, and $r_0^{-2} = \langle
r_i^{-2}\rangle$ is the equilibrium average of the inverse of the square of the
distance of an electron from an axis threading the torus. This result is valid
in three and two dimensions for arbitrary interactions, impurity potentials,
and magnetic fields.
