Bootstrap argument
= 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 $A(t)\leq C\varepsilon+C\int_0^t(1+s)^{-1}A(s)^2\,ds$ closes $A\leq2C\varepsilon$ when $\varepsilon\log(1+t)$ is sufficiently small. The strict improvement is what prevents a first failure time.