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AM-GM Inequality Proof

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You need the additional premise a, b ≥ 0, and then you can use a^(1/2) and b^(1/2) in your first inequality to get

((a^(1/2))^(2) + (b^(1/2))^(2))/2 ≥ a^(1/2)b^(1/2)

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