I'm reading S.Brendle's Ricci Flow and the Sphere Theorem. In page36 and page43, he says using Cor3.3, we can obtain some higher order covariant curvatures are uniformly bounded in $[0,T)$. But Cor3.3 just shows that $\sup\limits_{M}{|D^mR_{g(t)}|^2}$ is bounded in $[\frac{\tau}{2},\tau]$ ,if $\sup\limits_{M}{|R_{g(t)}|^2}<\tau^{-1} $, $\forall t \in[0,\tau]$. How to use the Cor3.3?
Detailed
Let M be a compact manifold, $g(t)$,$t \in [0,T)$,$T< \infty$ be a maximal solution to the Ricci flow on M. The curvature tensor of $g(t)$(means that $\sup\limits_{t\in[0,t)}\sup\limits_{M}|R_{g(t)}|<\infty$) is uniformly bounded for all t$\in[0,T)$.Show that $\sup\limits_{t\in[0,t)}\sup\limits_{M}{|D^mR_{g(t)}|}<\infty$