Claim:
$\lim_{k\to\infty} x[k]$ exist and if finite is $X(z)$ the Z-transform of $x[k]$ has no pole in $|z|>1$ and at most 1 pole at $z = 1$
Attempt: \begin{align*} X(z) &= \sum_{k\ge 0} x[k]z^{-k}\\\ H(z)/G(z) &= \sum_{k\ge 0} x[k]z^{-k}\\\ \end{align*}
First prove that no pole can be in $|z|>1$ \begin{align*} H(z)/G(z) &= \sum_{k\ge 0} x[k]z^{-k}\\\ H(z)/[(z-a)G'(z)] &= \sum_{k\ge 0} x[k]z^{-k}, a>1\\\ H(z)/[(z-a)G'(z)] &= x[0] + x[1]z^{-1} +x[2]z^{-2}+x[3]z^{-3}...\\\ H(z)/G'(z) &= (z-a)(x[0] + x[1]z^{-1} +x[2]z^{-2}+x[3]z^{-3}...)\\\ \end{align*}
Can someone help carry this further?