  Let $(M,g)$ be a compact connected spin manifold of dimension $n\geq 3$ whose
Yamabe invariant is positive. We assume that $(M,g)$ is locally conformally
flat or that $n \in \{3,4,5\}$. According to a positive mass theorem of Witten,
the constant term in the asymptotic development of the Green's function of the
conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the
sphere. In the present article, we will give a proof for this fact which is
considerably shorter than previous proofs. Our proof is a modification of
Witten's argument, but no analysis on asymtotically flat spaces is needed.
