Convergence rate of two sequences

Question

I've been reading a paper where the authors have proved linear convergence of the sequences $(x_k, y_k)$ to the optimal solutions $(x^\star, y^\star)$ by proving \begin{align} \frac{1}{a} \|x_k-x^\star\|^2 + a b \|y_k-y^\star\|^2 \leq \delta^k \left(\frac{1}{a} \|x_0-x^\star\|^2 + a b \|y_0-y^\star\|^2 + C\right)~ (1) \end{align} where $a >0$, $b>0$, $C >0$ and $\delta \in (0,1)$. My understanding is that $x_k$ converges linearly to $x^\star$ with rate $\delta$, i.e. \begin{align} \|x_k-x^\star\|^2 \leq \delta^k \left(\|x_0-x^\star\|^2 + C_0 \right) ~(2) \end{align} My question is giving Eq. (1), can I write Eq. (2)? Now, that I think of it, I think if I set $C_0 = a^2 b \|y_0-y^\star\|^2 + C$, then I will get Eq. (2).