In Evans's PDE, the section about Laplace's Equation, page 34 and page 36, I'm confusing about the following estimation: Given $U\subset\mathbb{R}^n$, $u(x)$ is harmonic in $U$, $v\in C^2(\bar{U})$, then
$$\left|\int_{\partial B(x,\varepsilon)} \dfrac{\partial v(y)}{\partial \nu} u(y) \,d S(y)\right|\leqslant C\varepsilon^{n-1}\max_{\partial B(x,\varepsilon)} |u|=o(1)$$
as $\varepsilon\to 0$.
I understand that the direct derivatives $\dfrac{\partial v(y)}{\partial \nu}$ can be take out of the integrand and become a constant, and the $\varepsilon^{n-1}$ comes from the surface area of $\partial B(x,\varepsilon)$.
However, why $C\varepsilon^{n-1}\max_{\partial B(x,\varepsilon)} |u|$ is $o(1)$? Maybe it follows from some properties of harmonic functions, I guess. But I'm not familar with this, could you please help me ?
I think $\epsilon^{n-1}\max_{\partial B(x,\epsilon)}|u|\leq\epsilon^{n-1}\max_{\overline{U}}|u|\rightarrow 0$.