Show that the pointwise limit of monotone increasing harmonic functions is harmonic

Let $\Omega \subseteq \mathbb{R}^n$ be open and consider a sequence $\{f_k\}_{k \in \mathbb{N}}$, $f \in C^2(\Omega)$ of harmonic functions in $\Omega$ such that $0 \le f_k \le f_{k+1}$ and for which $\displaystyle f(x) \equiv \sup_{k \in \mathbb {N}} f_k(x) < \infty$, each $x \in \Omega$. Prove that $f$ is harmonic in $\Omega$.