Proof of the energy estimates for the elliptic equations in Evans's Partial Differential Equations From PDE Evans, 2nd edition, page 318

THEOREM 2 (Energy estimates). There exist constants $\alpha, \beta > 0$ and $\gamma \ge 0$ such that $$|B[u,v]| \le \alpha \|u\|_{H_0^1(U)} \|v\|_{H_0^1(U)}$$ and $$\beta\|u\|_{H_0^1(U)}^2 \le B[u,u] + \gamma \|u\|_{L^2(U)}$$ for all $u,v \in H_0^1(U)$.
Proof. 1. We readily check 
  \begin{align*}
|B[u,v]| &\le \sum_{i,j=1}^n \|a^{ij}\|_{L^\infty} \int_U |Du| |Dv| \, dx \\
&\qquad + \sum_{I=1}^n \|b^i\|_{L^\infty} \int_U |Du| |v| \, dx + \|c\|_{L^\infty} \int_U |u| |v| \, dx \\
&\le \alpha \|u\|_{H_0^1(U)} \|v\|_{H_0^1(U)},
\end{align*}
  for some appropriate constant $\alpha$.

The proof continues on in the textbook but I am not reprinting the remaining steps of the proof here, because my question concerns only step 1 of the proof.
Question:
Why is the last inequality true? 
$$
 \sum_{i,j=1}^n \|a^{ij}\|_{L^\infty} \int_U |Du| |Dv| \, dx  + \sum_{I=1}^n \|b^i\|_{L^\infty} \int_U |Du| |v| \, dx + \|c\|_{L^\infty} \int_U |u| |v| \, dx \le \alpha \|u\|_{H_0^1(U)} \|v\|_{H_0^1(U)},
$$
I'm not sure, but I think it's because by definition $H_0^1=W_0^{1,1}$ (the closure of $C_c^\infty(U)$ in $H^1(U)=W^{1,1}(U)$) is the addition of all $u,v, Du,Dv \in L^p$. And the LHS of the inequality in question contains products of those elements. Again, I'm not entirely sure.
 A: First off, the notation $H^1_0(U)$ is shorthand for $W^{1,2}_0(U)$, i.e.
\begin{align*}
H^1_0(U) = \{ u \in L^2 : D u \in L^2(U), u|_{\partial U} = 0 \}, 
\end{align*}
with norm 
\begin{align*}
\| u \|_{H^1_0(U)}^2 = \| u \|_2^2 + \|Du \|^2_2.
\end{align*}
Here $D u$ is the weak derivative of $u$ and the boundary evaluation is interpreted in the trace sense. The bound in question follows from Cauchy-Schwarz
inequality.  In particular, 
\begin{align*}
\int_U  |Du||Dv|dx &\leq \left (\int_U |Du|^2 dx \right)^{1/2} \left (\int_U |Dv|^2 dx \right)^{1/2} \leq \| u \|_{H^1_0(U)} \| v \|_{H^1_0(U)}, \\
\int_U  |Du||v|dx &\leq \left (\int_U |Du|^2 dx \right)^{1/2} \left (\int_U |v|^2 dx \right)^{1/2} \leq \| u \|_{H^1_0(U)} \| v \|_{H^1_0(U)}, \\
\int_U  |u|||v|dx &\leq \left (\int_U |u|^2 dx \right)^{1/2} \left (\int_U |v|^2 dx \right)^{1/2} \leq \| u \|_{H^1_0(U)} \| v \|_{H^1_0(U)}.
\end{align*}
We then take 
\begin{align*}
\alpha = \sum_{i,j} \| a^{ij} \|_\infty + \sum_i \| b^i \|_\infty 
+ \| c \|_\infty.
\end{align*}
