= 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 $|y(t)|\leq B+c\max_{0\leq s\leq t}|y(s)|^2$, where $B+cR^2<R$. At a first time $T$ with $|y(T)|=R$, the estimate gives $R\leq B+cR^2<R$, a contradiction. Hence $|y(t)|<R$ for every finite time. A separate uniform estimate is needed to conclude $\sup_{t\geq0}|y(t)|<R$, since pointwise strict inequalities alone allow equality of the supremum.
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