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I was recently reading up on the Price Elasticity of Demand and the section of Inelastic demand mentions that if a firm has inelastic demand for its product and wishes to increase its total revenue then it should raise the price of the product. The author has done one calculation to somehow convince the reader, but I think that anyone can conjure numbers just for the sake of argument. I am looking for a mathematical argument that will be rigorous enough to convince me. Please note that I get the intuition behind why this might happen but my textbook does not provide any mathematical reasoning as to why this must happen. Any insights will be much appreciated.