Is this correct? - showing analytic function is constant.

for analytic function f:A-->C, where A is an open connected set. If f(A) is completely contained in the line {x+iy | x=2y}, show that f is constant.

My answer: i said that the analytic function f(x,y)=u(x,y)+iv(x,y) has to be in the form u(x,y)=x, v(x,y)=2x. Then Cauchy-Riemann relations:

du/dx=1 =/= 0=dv/dy

du/dy=0 =/= -2=-dv/dx

So the only way they can hold is if x=c, c some constant, so that all the partial derivatives are equal to 0.

Is this correct? (I feel like its a little too simple...)