Understanding the (Partial) Converse to Cauchy-Riemann We have that for a function $f$ defined on some open subset $U  \subset \mathbb{C}$ then the following if true:

Suppose $u=\mathrm{Re}(f), v=\mathrm{Im}(f)$ and that all partial derivatives $u_x,u_y,v_x,v_y$ exists and are continuous on $U$. Suppose further that they satisfy the Cauchy-Riemann equations. Then $f$ is holomorphic on $U$.

The proof for this is readily available, though there is a subtlety that I can't understand.
We essentially want to compute $\lim_{h \rightarrow 0} \dfrac{f(z+h)-f(z)}{h}$ where $h=p+qi \in \mathbb{C}$.
We need a relationship like $u(x+p,y+q)-u(x,y)=pu_x(x,y)+qu_y(x,y)+o(|p|+|q|)$. Why is this relationship true?

Denote $h=\pmatrix{p\\q}.$ 
Since $u(x,\ y)$ is differentiable at the point $(x,\ y),$   increment of $u$ can be represented as
 $$u(x+p,\ y+q)-u(x,\ y)=Du\;h+o(\Vert h\Vert)=pu_x(x,\ y)+qu_y(x,\ y)+o(|p|+|q|),$$
where $Du=\left(\dfrac{\partial{u}}{\partial{x}}, \ \dfrac{\partial{u}}{\partial{y}} \right).$
