Absolutely convergent series of complex functions.

I have to do the following exercise:

To let $ {f_n (z) } _ {n in mathbb {N}} $ a sequence of complex functions and let $ sum_ {n = 1} ^ infty f_n (z) $,

Proof of this: if $ sum_ {n = 1} ^ infty | f_n (z) | $ then converges $ sum_ {n = 1} ^ infty f_n (z) $ converges.

I know how to prove it for a series $ sum_ {n = 1} ^ infty z_n $ with complex numbers $ z_n = x_n + iy_n $ because when $ sum_ {n = 1} ^ infty | z_n | $ converges, you can watch that $ | x_n | <| z_n | $ and $ | y_n | <| z_n | $ then by the comparison criteria the real number series $ sum_ {n = 1} ^ infty | x_n | $ and $ sum_ {n = 1} ^ infty | y_n | $ converge and we know for real series that implies that $ sum_ {n = 1} ^ infty x_n $ and $ sum_ {n = 1} ^ infty y_n $ converge.

When we call $ R_n = sum_ {k = 1} ^ n x_n $, $ I_n = sum_ {k = 1} ^ ny_n $ and $ S_n = sum_ {k = 1} ^ n z_n $,

And $ lim_ {n rightarrow infty} R_n = x $, $ lim_ {n rightarrow infty} I_n = y $, then

$$ lim_ {n rightarrow infty} S_n = lim_ {n rightarrow infty} R_n + i lim_ {n rightarrow infty} I_n = x + iy. $$

Then $ S_n $ converges and $ sum_ {n = 1} ^ infty z_n $ is also good

Is it enough to call? $ {w_n } = {f_n (z) } $ in my original problem and just apply this proof?

Thank you in advance.