Source: Evans' PDE book, section 7.1, near equation (48)
In step 2, we have $$ \begin{align} \|\boldsymbol{u}_m'\|_{L^2(U)}^2 + & \frac{d}{dt}\biggl(\frac12 A[\boldsymbol{u}_m,\boldsymbol{u}_m]\biggr) \\ & \leq \frac{C}{\epsilon} \Bigl(\|\boldsymbol{u}_m\|_{H_0^1(U)}^2 + \|\boldsymbol{f}\|_{L^2(U)}^2\Bigr) + 2\epsilon \|\boldsymbol{u}_m'\|_{L^2(U)}^2. \end{align} $$
Choosing $\epsilon=1/4$ we have $$ \|\boldsymbol{u}_m'\|_{L^2(U)}^2 + \frac{d}{dt}A[\boldsymbol{u}_m,\boldsymbol{u}_m] \leq C \Bigl(\|\boldsymbol{u}_m\|_{H_0^1(U)}^2 + \|\boldsymbol{f}\|_{L^2(U)}^2\Bigr). $$
Integrate from $0$ to $T$ to get $$ \int_0^T \|\boldsymbol{u}_m'\|_{L^2(U)}^2\,dt + A[\boldsymbol{u}_m(T),\boldsymbol{u}_m(T)] \leq C\biggl(A[\boldsymbol{u}_m(0),\boldsymbol{u}_m(0)] + \int_0^T \|\boldsymbol{u}_m\|_{H_0^1(U)}^2 + \|\boldsymbol{f}\|_{L^2(U)}^2\,dt\biggr). $$
How can we obtain the desired inequality in the book with $A[\boldsymbol{u}_m(T),\boldsymbol{u}_m(T)]$ being replaced by $\sup_{0\leq t\leq T} A[\boldsymbol{u}_m(t),\boldsymbol{u}_m(t)]$?