I am trying to solve this exercise :
Consider $U$ open , bounded set in $R^n$ with smooth boundary. Let $u \in C^2(U)$ a harmonic function. Suppose that for each $x \in \partial U $ there exists a sequence $a_k$ in $U$ with
$$ a_k \rightarrow x$$ and
$$ \limsup \ u(a_k) \leq M$$
where $M$ is a constant independent of $x$. Show that $u \leq M $ in $U$.
To prove this exercise I am trying to use the maximum principle :
Consider $U$ an open, bounded set in $R^n$ with smooth boundary. Let $u \in C^2(U) \cap C(\overline{U})$ a harmonic function. Then the maximum and minimum occurs in the boundary of $U$. Someone can give me a hand to prove the exercise? Any help is appreciated.
Thanks in advance!