Real Inverse Hyperbolic Cosine is Strictly Increasing

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Theorem

The real inverse hyperbolic cosine function is strictly increasing, that is:

$\forall x, y \ge 1 : x < y \implies \cosh^{-1} x < \cosh^{-1} y$


Proof

\((1):\quad\) \(\displaystyle x\) \(<\) \(\displaystyle y\) Assumption
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2\) \(<\) \(\displaystyle y^2\) Axiom $\R \text O 2$: Usual ordering is compatible with multiplication
\(\displaystyle \leadsto \ \ \) \(\displaystyle x^2 - 1\) \(<\) \(\displaystyle y^2 - 1\)
\((2):\quad\) \(\displaystyle \leadsto \ \ \) \(\displaystyle \sqrt {x^2 - 1}\) \(<\) \(\displaystyle \sqrt {y^2 - 1}\) Square Root is Strictly Increasing
\(\displaystyle \leadsto \ \ \) \(\displaystyle x + \sqrt {x^2 - 1}\) \(<\) \(\displaystyle y + \sqrt {y^2 - 1}\) $(1) + (2)$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cosh^{-1} x\) \(<\) \(\displaystyle \cosh^{-1} y\) Definition of Real Inverse Hyperbolic Cosine

$\blacksquare$