Let f be holomorphic in an open set Ω ⊂ $\mathbb{C}$. Show that if u = Re f and v = Im f then u and v satisfy the Cauchy-Riemann equations. Show further that u is harmonic.
[You may assume without proof that u and v are twice-continuously differentiable in Ω.]
I cannot show that u and v satisfy the C-R equations (please help) but for the 2nd part,
From the Cauchy–Riemann equations,
$$u_{xx} = −v_{yx} = −v_{xy} = −u_{yy}$$
The first equality is the CR equation $u_{x}$ = $v_{y}$, the second is the symmetry of mixed partial derivatives, and the third is the CR equation $-v_{x}$ = $u_{y}$. Thus the continuity of the second partials is needed for $v_{xy}$ = $v_{yx}$ and hence for the proof to work.
Is this right?
Your argument for the second part looks fine, but you could get rid of the last sentence. For the first part, use the limit definition $$f'(z) = \lim_{h\to 0} \frac{f(z + h) - f(z)}{h}$$ and consider $h\to 0$ along the real and imaginary axes separately. In one case, $f'(z) = u_x + iv_x$, and in the other case, $f'(z) = -i(u_y + iv_y) = v_y - iu_y$. Compare real and imaginary parts to establish the Cauchy-Riemann equations.