Problem
1) Prove that if the real and imaginary part of a holomorphic function are of class $C^2$, then they are harmonic.
2)Deduce from 1) that if $u(x,y) \in C^2$ is a function that admits a harmonic conjugate, then $u$ is harmonic.
My attempt
I could prove part 1) using the Cauchy-Riemann equations. I am having problems with 2): I am not suppose to use that holomorphic functions are analytic (which also means $u,v \in C^{\infty}$).
I want to show that $u_{xx}+u_{yy}=0.$
By hypothesis, $u$ is the real part of a holomorphic function $f(x+iy)=u(x,y)+iv(x,y).$
Using the Cauchy-Riemann equations, I have $$u_{xx}=v_{xy} \space;\space u_{yy}=-v_{yx}$$
I don't know how one can deduce from here that $\Delta u=0$. I would appreciate some help.