Proof of Z-Transform Property | X(z) and x(n) Relation

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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?