We have the inequality αn(t)≤4π(3π)1/3exp{∫t0(1+3FP(σ))dσ}⋅∫t0P(σ)2(3CD21)1/3αn−1(σ)dσ for n=2,3,…. (We notice that αn appears on both sides of the inequality.)
Why does it follow that the infinite series ∞∑n=1αn(t) converges locally uniformly on R+0 (that is, converges on [0,T] for 0≤t≤T)?
This comes from page 354 of the journal that contains the paper "Global symmetric solutions of the initial value problem of stellar dynamics" by Jurgen Batt.