  Quantum inequalities are constraints on how negative the weighted average of
the renormalized stress-energy tensor of a quantum field can be. A
null-projected quantum inequality can be used to prove the averaged null energy
condition (ANEC), which would then rule out exotic phenomena such as wormholes
and time machines. In this work we derive such an inequality for a massless
minimally coupled scalar field, working to first order of the Riemann tensor
and its derivatives. We then use this inequality to prove ANEC on achronal
geodesics in a curved background that obeys the null convergence condition.
