[SOLVED] Complex Analysis Proof

Write $f(x+iy) = u(x,y) + iv(x,y)$.
We know that $v(x,y)=0$ by hypothesis.
But, $u_x = v_y \implies u_x = 0 \implies u(x,y) = g(y)$.
And, $u_y = -v_x \implies u_y = 0 \implies u(x,y) = h(x)$.
Therefore, $g,h$ need to be constant functions.
And so $u(x,y) = c$.