The real part of a holomorphic function is harmonic

Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be holomorphic on $U$. Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$, as real functions of $(x, y)$ on $U$, with $z = x + iy$. Then $u$ and $v$ are twice continuously differentiable, and $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \qquad \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$$ on $U$. Thus $u$ and $v$ are harmonic functions on $U$.