  In this paper, we study the hydrostatic approximation for the Navier-Stokes
system in a thin domain. When the convex initial data with Gevrey regularity of
optimal index 3/2 in x variable and Sobolev regularity in y variable, we
justify the limit from the anisotropic Navier-Stokes system to the hydrostatic
Navier-Stokes/Prandtl system. Due to our method in the paper is independent of
{\epsilon}, by the same argument, we also get the hydrostatic
Navier-Stokes/Prandtl system is well-posedness in the optimal Gevrey space. Our
results improve the Gevrey index in [14, 34] whose Gevrey index is 9/8 .
