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Theorem: Inequality of the Arithmetic Mean
From the order relation of two real numbers $x < y$ it follows for the arithmetic mean $\frac{x+y}{2}$ that in lies in between of $x$ and $y$, formally $$x < \frac{x+y}{2} < y.$$
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References
Bibliography
 Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition