  We present a calculation of the low energy Greens function in $1+\epsilon$
dimensions using the method of extended poor man's scaling, developed here. We
compute the wave function renormalization $Z(\omega)$ and also the decay rate
near the Fermi energy. Despite the lack of $\omega^2$ damping characteristic of
3-dimensional Fermi liquids, we show that quasiparticles do exist in
$1+\epsilon$ dimensions, in the sense that the quasiparticle weight $Z$ is
finite and that the damping rate is smaller than the energy. We explicitly
compute the crossover from this behavior to a 1-dimensional type
Tomonaga-Luttinger liquid behavior at higher energies.
