First-exit argument

A first-exit argument proves preservation of a bound by contradiction at the earliest time the bound could fail. Continuity of the evolving quantity ensures that, if a trajectory starts inside a closed region and later leaves it, there is a first contact with its boundary. An estimate valid up to that contact can then rule the contact out. This is useful before a global supremum norm is known to be finite.

For example, suppose a continuous $y$ with $|y(0)|<R$ satisfies $|y(t)|\le B+c{\max }_{0\le s\le t}|y(s){|}^2$, where $B+c{R}^2<R$. At a first time $T$ with $|y(T)|=R$, the estimate gives $R\le B+c{R}^2<R$, a contradiction. Hence $|y(t)|<R$ for every finite time. A separate uniform estimate is needed to conclude ${\sup }_{t\ge 0}|y(t)|<R$, since pointwise strict inequalities alone allow equality of the supremum.