Show that exist a function $u$ continuous in $\overline{\Omega}$: $$\lim_{k\rightarrow \infty}u_k=u\quad \text{uniformly on }\overline{\Omega}$$.
The problem is: Let $\Omega\subset \mathbb R^n$ bounded set. Let be $g$ a continuous function over $\partial\Omega$ and $\{u_k\}$ continuous function over $\overline{\Omega}$ of class $C^2$ and harmonic functions such that: $\lim_{k\rightarrow \infty}u_k|_{\partial\Omega}=g,\quad\text{uniformly on}\quad\partial\Omega$.\
My attempt is, all function $u_k$ satisfies that $\Delta u_k=0$ and as $u_k$ is continuous in $\overline{\Omega}$ compact set (because is bounded and closed in $\mathbb R^n$) so its maximum and minimum is reached in $\overline{\Omega}$. The other side
the limit, $\lim_{k\rightarrow \infty}u_k$ exist because is limit of the functions continuous in one compact set $\overline{\Omega}$. My problem star here, if i call $\tilde{g}:=\lim_{k\rightarrow \infty}u_k$ then i want to show that $g$ is a continuous function, and your restriction over $\partial\Omega$ is too.
Perhaps i have to use the principle of Maximo for the function $g=\lim u_k$ and i remember that limit uniform of continuous functions over a compact set implies that the limit function is continuous, but i would like help or hints please, thank you!!!.
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