  We consider the stress tensor in the massless Sine--Gordon model in the
finite regime $\beta^2 < 4 \pi$ of the theory. We prove convergence of the
renormalised perturbative series for the interacting stress tensor defined
using the Bogoliubov formula in an arbitrary Hadamard state, even for the case
that the smearing is only along a one-dimensional time-like worldline and not
in space-time. We then show that the interacting energy density, as seen by an
observer following this worldline, satisfies an absolute lower bound, that is a
bound independent of the quantum state. Our proof employs and generalises
existing techniques developed for free theories by Flanagan, Fewster and Smith.
