  The aim of this paper is to give a constructive proof of one of the basic
theorems of tropical geometry: given a point on a tropical variety (defined
using initial ideals), there exists a Puiseux-valued ``lift'' of this point in
the algebraic variety. This theorem is so fundamental because it justifies why
a tropical variety (defined combinatorially using initial ideals) carries
information about algebraic varieties: it is the image of an algebraic variety
over the Puiseux series under the valuation map. We have implemented the
``lifting algorithm'' using Singular and Gfan if the base field are the
rational numbers. As a byproduct we get an algorithm to compute the Puiseux
expansion of a space curve singularity in (K^{n+1},0).
