Euler Number and AM GM Inequality - Stories in Real Analysis

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A proof toward the existence of Euler number; an application of Arithmetic Mean - Geometric Mean Inequality

Problem useful for I.S.I B.Stat B.Math Entrance, CMI Entrance and Math Olympiad
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For all this to hold true, n must be and integer. How would we show then (1+1/3.1)^3.1>(1+1/3)^3. I think this a faulty proof. Please correct me if I am wrong.

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