Sequences in complex (just a clarifying question)

[bennyska]

Homework Statement

(excuse lack of latex)
show that if SUM(z_n)= S and SUM(w_n= T, then SUM(z_n + w_n) = S + T

Homework Equations

The Attempt at a Solution

so if I'm doing this right, this is pretty easy, i think. they want me to use a theorem that says if z_n=x_n +iy_n, and SUM(z_n)= S, where S = X + iY, this happens iff SUM(x_n)= X and SUM(y_n)= Y.

so my question is am i allowed to say SUM(z_n + w_n) = SUM(z_n) + SUM(w_n) = SUM(x_n) + iSUM(y_n) + SUM(u_n) + iSUM(v_n)? I'm not sure if I'm violating anything by breaking it up like this. it all follows very quickly from here.

*edit: forgot to erase double standard questions after preview

 

[Office_Shredder]

You can break it up as:
SUM(x_n+u_n)+iSUM(y_n+v_n)

just using the property that you posted (make sure you understand why). To go further you need to do something else