Question about the proof of the Rellich-Kondrachov theorem in "Partial Differential equations" by C. Evans.

Question

Im trying to understand the proof of the Rellich-Kondrachov theorem in C. Evans book "Partial Differential equations", that is, theorem 1 in section 5.7. After proving that $u_m^\epsilon \rightarrow u_m$ in $L^q(V)$ he proves that $\{u_m^\epsilon\}_{m \geq 1}$ is uniformly bounded and equicontinuous for fixed $\epsilon > 0$. He then estimates $\lvert u_m^\epsilon(x)\rvert$ which obviously proves that $\{u_m^\epsilon\}_{m \geq 1}$ is uniformly bounded, but then he estimates $\lvert Du_m^\epsilon(x)\lvert$, which is the euclidean norm of the gradient vector, and claims that this shows uniform equicontinuity, without further explanation. My question is how does uniform equicontinuity follows from these two estimates?