$\{u_{n}\}$ harmonic and converging uniformly to $u \Rightarrow $ $u$ harmonic

Question

Let $A$ be an open set, $\{u_{n}\}$ a sequence of harmonic functions on $A$, converging to $u$ uniformly on compact subsets of $A$ . Then $u$ is harmonic on $A$.

Any hint ?

Answer 1

We can use Gauss Mean Value Theorem for Harmonic Functions: A function $u :\Omega\to\mathbb R$ is harmonic if and only if $$ u(x_0)=\frac{1}{|B_r|}\int_{B_r(x_0)}u(x_0+h)\,dh $$ for every $\overline{B}_r(x_0)\subset\Omega$.

Then uniform convergence passes is both sides of the above.