If $f$ and $g$ both satisfy Cauchy Riemann equations at point $z$, prove $f+g$ and $fg$ satisfy the Cauchy Riemann equations

Question

So if $f=u+iv$, and $g=m+in$. Trying to prove $fg$ follows the Cauchy Riemann equations: I got $fg=(mu-nv)+i(nu+mv)$, letting $a=mu-nv$ and letting $b=nu+mv$, I differentiated $a$ and $b$, by $x$ and $y$ but I can't equate the differentials so I don't know what to do.

Answer 1

Note that\begin{align}\frac{\partial(mu-nv)}{\partial x}&=\frac{\partial m}{\partial x}u+\frac{\partial u}{\partial x}m-\frac{\partial n}{\partial x}v-\frac{\partial v}{\partial x}n\\&=\frac{\partial n}{\partial y}u+\frac{\partial v}{\partial y}m+\frac{\partial m}{\partial y}v+\frac{\partial u}{\partial y}n\\&=\frac{\partial n}{\partial y}u+\frac{\partial u}{\partial y}n+\frac{\partial v}{\partial y}m+\frac{\partial m}{\partial y}v\\&=\frac{\partial(nu+mv)}{\partial y}.\end{align}Can you take it from here?