There are two ways to define a tropical variety; one is to define it as the image under the order map of a variety over the Puiseux-series, and the other one is more combinatorial. We present a constructive proof that the two definitions coincide, i.e. that each point in the combinatorially-defined tropical variety can be lifted to a Puiseux-valued point. This shows that tropical geometry deals with combinatorial objects which allow conclusions about algebraic varieties, because they are images of algebraic varieties at the same time.