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Set $a=1$ in the parametric solution. With $b=1-\Omega$,ConsequentlyThe endpoint values are obtained by limits. A particularly direct proof of the age bound avoids subtracting nearly equal terms: from the Friedmann equation,For nonnegative component densities, $0\le \Omega \le 1$ and $a\le \Omega +(1-\Omega )a\le 1$. The integrand lies between $\sqrt{a}$ and $1$. Integrating therefore givesEquality holds for pure matter and pure coasting fluid, respectively. These are the age of a flat matter-coasting-fluid universe and the coasting-fluid age bound; the positivity range is necessary for the stated universal inequality.

Normalize the present scale factor to one. For $0<\Omega <1$, the parametric solution givesThus the age of a flat matter-coasting-fluid universe isA direct age integral proves the bound more transparently than manipulating inverse hyperbolic functions. Since $\dot{a}=H_0\sqrt{\Omega /a+b}$,For $0\le a\le 1$ and $0\le \Omega \le 1$, one has $a\le \Omega +ba\le 1$, henceIntegrating givesEquality on the left is the pure pressureless matter universe, $\Omega =1$; equality on the right is the pure coasting fluid, $\Omega =0$. These are all physically allowed density fractions for the nonnegative two-fluid model. The integrand decreases with $\Omega$, so the coasting-fluid age bound is approached continuously between those endpoints. In the closed expression the apparent $b=0$ singularities cancel using $\mathrm{arsinh}x=x-x^3/6+\cdots$; as $\Omega \to 0$, the term $\Omega \log (1/\Omega )$ vanishes. The limits are respectively $H_0{t}_0=2/3$ and $1$.