I'm trying to show that $y[n]=x[n]*h[n]$ turns into $Y(z) = X(z)H(z)$ in Z-domain by first applying convolution then by taking the inverse Z-transform of the $Y(z)$, stating that it's the same sequence after all. However, in power series expansion an extra term appears which kills the equation. You can find the question below; I added it as an image.
I'd be grateful if you could help. Thank you