Real Area Hyperbolic Cosine is Strictly Increasing
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Theorem
The real area hyperbolic cosine function is strictly increasing, that is:
- $\forall x, y \ge 1 : x < y \implies \arcosh x < \arcosh y$
Proof
| \(\text {(1)}: \quad\) | \(\ds x\) | \(<\) | \(\ds y\) | Assumption | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x^2\) | \(<\) | \(\ds y^2\) | Real Number Axiom $\R \text O2$: Usual Ordering is Compatible with Multiplication | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x^2 - 1\) | \(<\) | \(\ds y^2 - 1\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \sqrt {x^2 - 1}\) | \(<\) | \(\ds \sqrt {y^2 - 1}\) | Square Root is Strictly Increasing | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds x + \sqrt {x^2 - 1}\) | \(<\) | \(\ds y + \sqrt {y^2 - 1}\) | $(1) + (2)$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \arcosh x\) | \(<\) | \(\ds \arcosh y\) | Definition of Real Inverse Hyperbolic Cosine |
$\blacksquare$