Monotone Jeans growth with positive cosmological constant

If ${\eta }^2=-s^2<0$, $\delta (0)>0$ and $\dot{\delta }(0)\ge 0$, the integrating factor identity $(e^{2\mu t}\dot{\delta }{)}^{\prime }=s^2{e}^{-\mu t}\delta$ proves that $\delta$ stays positive and increases. Its finite limit satisfies ${\delta }_{\infty }=\delta (0)+\dot{\delta }(0)/(2\mu )+s^2{\int }_0^{\infty }{e}^{-3\mu t}\delta (t)dt/(2\mu )>\delta (0)$. The initial sign condition is essential: the positive decaying modified Bessel function mode has limit zero.