To get a small strip near the boundary of a smooth bounded domain

Question

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ and $u:\Omega\to\mathbb{R}$ be a non-negative function such that $lim_{x\to x_0}u(x)=0$ for every $x_0\in\partial\Omega$ where $x\in\Omega$. Then for any $\epsilon>0$ there exists $\delta>0$ such that $u\leq\frac{\epsilon}{2}$ in $A_{\delta}=\{x\in\overline{\Omega}:dist(x,\partial\Omega)<\delta\}$. Can you please explain whether this is true or not. Please give an explanation. Thank you very much in advance.