Bootstrap argument

A bootstrap argument assumes an a priori bound on an initial time interval and derives a strict improvement using an energy estimate or another inequality. Continuity extends the improved bound to the entire interval under consideration. For example $A(t)\le C\epsilon +C{\int }_0^t(1+s{)}^{-1}A(s{)}^{2}ds$ closes $A\le 2C\epsilon$ when $\epsilon \log (1+t)$ is sufficiently small. The strict improvement is what prevents a first failure time.