# Mean of Unequal Real Numbers is Between them

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## Theorem

- $\forall x, y \in \R: x < y \implies x < \dfrac {x + y} 2 < y$

## Proof

First note that:

\(\displaystyle 0\) | \(<\) | \(\displaystyle 1\) | Real Zero is Less than Real One | ||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle 0 + 0\) | \(<\) | \(\displaystyle 1 + 1\) | Real Number Inequalities can be Added | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle 0\) | \(<\) | \(\displaystyle \frac 1 {1 + 1}\) | Reciprocal of Strictly Positive Real Number is Strictly Positive | |||||||||

\((1):\quad\) | \(\displaystyle \implies \ \ \) | \(\displaystyle 0\) | \(<\) | \(\displaystyle \frac 1 2\) |

Then:

\(\displaystyle x\) | \(<\) | \(\displaystyle y\) | |||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle x + x\) | \(<\) | \(\displaystyle x + y\) | Real Number Axioms: $\R O1$: compatibility with addition | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \left({x + x}\right) \times \frac 1 2\) | \(<\) | \(\displaystyle \left({x + y}\right) \times \frac 1 2\) | Real Number Axioms: $\R O2$: compatibility with multiplication and from $(1)$ | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle x\) | \(<\) | \(\displaystyle \frac {x + y} 2\) | Definition of Division |

Similarly:

\(\displaystyle x\) | \(<\) | \(\displaystyle y\) | |||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle x + y\) | \(<\) | \(\displaystyle y + y\) | Real Number Axioms: $\R O1$: compatibility with addition | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \left({x + y}\right) \times \frac 1 2\) | \(<\) | \(\displaystyle \left({y + y}\right) \times \frac 1 2\) | Real Number Axioms: $\R O2$: compatibility with multiplication and from $(1)$ | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \frac {x + y} 2\) | \(<\) | \(\displaystyle y\) | Definition of Division |

$\blacksquare$

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $2 \ \text{(k)}$