When I read the strong maximum principle in chapter 6 of Evans' book , I meet some problems.
Theorem 3 (Strong maximum principle) . Assume $u\in C^2(U)\cap C(\overline{U})$ and $c\equiv 0 $ in $U$. Suppose also $U$ is connected , open and bounded.
(i) If $$Lu\leq0$$ in $U$ and $u$ attains its maximum over $\overline{U}$ at an interior point , then $u$ is constant within $U$.
(ii) $\cdots$
Proof. Write $M:=\max_{\overline{U}}u$ and $C:=\{x\in U|u(x)=M\}$. Then if $u\not\equiv M$ , set $V:=\{x\in U|u(x)<M\}.$ Choose a point $y\in V$ satisfying $dist(y,C)<dist(y,\partial U)$ , $\cdots$
My doubts: Why could we choose such a point $y$ ? Could someone give me some advice ? I don't know how to deal with this.