Lemma 3.2.5 (Gronwall).label Let $f \in C([0, T]; \real)$, and $c, K > 0$ such that
\[f(t) \le c + K\int_{0}^{t} f(s)ds\]
for all $t \in [0, T]$, then
\[\frac{d}{dt}\braks{e^{-Kt}\int_0^t f(s)ds}\le ce^{-Kt}\]
and $f(t) \le ce^{Kt}$.
Lemma 3.2.5 (Gronwall).label Let $f \in C([0, T]; \real)$, and $c, K > 0$ such that
for all $t \in [0, T]$, then
and $f(t) \le ce^{Kt}$.