Please check if I did this proof right.
Show that if $\phi (x,y)$ is harmonic, then $\phi_{x} - i\phi_{y}$ is analytic. ( You may assume that $\phi$ has continuous partial derivatives of all orders.)
$\underline {Proof}$:
Let $\phi$ be harmonic and $\phi$ has continuous partial derivatives.
Now if we consider $u =\phi$ and $v = \phi$, it is enough to show that $u$ and $v$ satisfies the Cauchy-Riemann equations
$$\frac{\partial u}{\partial v} = \phi_{xx} = -\phi_{yy} = \frac{\partial v}{\partial y}$$
$$\frac{\partial u}{\partial y} = \phi_{xy} = -\frac{\partial v}{\partial x}$$
So then $u = \phi_{x}$ and $v = \phi_{y}$, satisfies the Cauchy-Riemann equation and hence $\phi{x} - \phi{y}$ is analytic.
This looks good except that you have quite a few typos. You presumably mean to say $u=\phi_x$ and $v=-\phi_y$, and $\frac{\partial u}{\partial x}$ instead of $\frac{\partial u}{\partial v}$. And at the end, you presumably mean $u+iv=\phi_x-i\phi_y$ rather than $\phi x-\phi y$. You could also point out explicitly where you are using the fact that $\phi$ is harmonic, since this is kind of implicitly buried in one of your equations.