Proposition 1.4: If x and y are real numbers, then 2xy (less than or equal to) x

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Proposition 1.4: If x and y are real numbers, then 2xy (less than or equal to) x^2 + y^2 and xy (less than or equal to) (x+y)^2 /(2^2). If x and y are also non negative, then (xy)^(1/2) (less than or equal to) (x+y)/2. Equality holds in each only when x=y.

Proof: We begin with 0 (less than or equal to) (x-y)^2 = x^2 - 2xy + y^2 and observe that equality holds only when x=y. Adding 2xy yields 2xy (less than or equal to) x^2 + y^2. Adding another 2xy yields 4xy (less than or equal to) x^2 + 2xy + y^2 = (x + y)^2, which we divide by 4 to obtain xy (less than or equal to) (x+y)^2 / (2^2). If x (greater than or equal to) 0 and y (greater than or equal to) 0, then also xy (greater than or equal to) 0, and we can take positive square roots in xy (less than or equal to) (x+y)^2 / (2^2).

Propostion1.1 yields:

xy (less than or equal to) (x+y) / 2

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