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This paper is a mechanism design study for a monopolist selling multiple identical items to potential buyers arriving over time. Participants in our model are time sensitive, with the same discount factor; potential buyers have unit demand and arrive sequentially according to a renewal process; and valuations are drawn independently from the same regular distribution. Invoking the revelation principle, we restrict our attention to direct dynamic mechanisms taking a sequence of valuations and arrival epochs as input. We define two properties (discreteness and stability), and prove under further distributional assumptions that we may at no cost of generality consider only mechanisms satisfying them. This effectively reduces the mechanism input to a sequence of valuations and leads to formulate the problem as a dynamic program (DP). As this DP is equivalent to a well-known infinite-horizon asset-selling problem, we finally characterize the optimal mechanism as a sequence of posted prices increasing with each sale. Remarkably, this result rationalizes somewhat the frequent restriction to dynamic pricing policies and impatient buyers assumption. Our numerical study indicates that, under various valuation distributions, the benefit of dynamic pricing over a fixed posted price may be small. Besides, posted prices are preferable to online auctions for a large number of items or high interest rate, but in other cases auctions are close to optimal and significantly more robust.