  The precision of radial velocity (RV) measurements to detect indirectly
planetary companions of nearby stars has improved to enable the discovery of
extrasolar planets in the Neptune and Super-Earth mass range. Discoveries of
Earth-like planets by means of ground-based RV programs will help to determine
the parameter Eta_Earth, the frequency of potentially habitable planets around
other stars. In search of low-mass planetary companions we monitored Proxima
Centauri (M5V) as part of our M dwarf program. In the absence of a significant
detection, we use these data to demonstrate the general capability of the RV
method in finding terrestrial planets. For late M dwarfs the classic liquid
surface water habitable zone (HZ) is located close to the star, in which
circumstances the RV method is most effective. We want to demonstrate that late
M dwarfs are ideal targets for the search of terrestrial planets with the RV
technique. We obtained differential RV measurements of Proxima Cen over a time
span of 7 years with the UVES spectrograph at the ESO VLT. We determine upper
limits to the masses of companions in circular orbits by means of numerical
simulations. The RV data of Proxima Cen have a total rms scatter of 3.1 m/s and
a period search does not reveal any significant signals. As a result of our
companion limit calculations, we find that we successfully recover all test
signals with RV amplitudes corresponding to planets with m sin i > 2 - 3
M_Earth residing inside the HZ of Proxima Cen with a statistical significance
of >99%. Over the same period range, we can recover 50% of the test planets
with masses of m sin i > 1.5 - 2.5 M_Earth. Based on our simulations, we
exclude the presence of any planet in a circular orbit with m sin i > 1
M_Neptune at separations of a < 1 AU.
