Proving Z{x*(n)} = X*(z*) for complex signals

[SpaceDomain]

Homework Statement

Using the definition of the z-transform, show that if $$X(z)$$ is the z-transform of $$x(n) = x_{R}(n) +jx_{I}(n)$$, then:
$$Z\{x^{*}(n)\}=X^{*}(z^{*})$$

Homework Equations

z-tranform definition:

$$Z\{x(n)\}=X(z)=\sum x(n)z^{-n}$$

The Attempt at a Solution

$$x(n) = x_{R}(n) + jx_{I}(n) \Longrightarrow x^{*}(n) = x_{R}(n) - jx_{I}(n)$$

$$Z\{x^{*}(n)\}=Z\{x_{R}(n) - jx_{I}(n)\}=\sum x^{*}(n)z^{-n}$$

$$=\sum [x_{R}(n) - jx_{I}(n)]z^{-n}$$

$$=\sum [x_{R}(n)z^{-n} - jx_{I}(n)z^{-n}]$$$$=\sum x_{R}(n)z^{-n} - j \sum x_{I}(n)z^{-n}$$

 

[SpaceDomain]

This is where I am stuck. Am I going in the right direction?