In this paper we present new results on the local and global convergence property of solutions to an optimization model where the objective function is a discounted sum of stationary one-period utilities. The asymptotic local turnpike is given without differentiability assumptions but imposing some mild curvature restrictions on the utility function. This approach allows us to get easy estimates on the range of discount factors and the size of the neighborhood for which the asymptotic property occurs. The paper concludes by providing two global turnpike theorems. The first one is an asymptotic theorem derived from a result similar to Scheinkman's visit lemma. The second one turns out to be a restatement of McKenzie's neighborhood turnpike theorem.