# Real Inverse Hyperbolic Cosine is Strictly Increasing

$\forall x, y \ge 1 : x < y \implies \cosh^{-1} x < \cosh^{-1} y$
 $(1):\quad$ $\displaystyle x$ $<$ $\displaystyle y$ 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
$\blacksquare$