\left( C_f + \lambda_k C_h \right).
    \end{aligned}$$ Combining this with , we will obtain $$\begin{aligned}
\label{incrementalineq2}
    \left \|x_{k+1} - x_{\lambda_k}^* \right \|^2          &\leq \left( 1- \frac{\gamma_k \lambda_k \mu_h}{m} \right) \left \|x_{k} - x_{\lambda_k}^* \right \|^2 \nonumber \\ &+ 6 m^2 \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2).
    \end{aligned}$$ Next, we relate $x_k$ to $x_{\lambda_{k-1}}^*$. We have $$\begin{aligned}
    &\left \|x_{k} - x_{\lambda_k}^* \right \|^2=\left \|x_{k} - x_{\lambda_{k-1}}^* \right\|^2 + \left \|x_{\lambda_k}^* - x_{\lambda_{k-1}}^* \right\|^2 \\ & + \underbrace{2 \left(x_{k} - x_{\lambda_{k-1}}^* \right)^T \left(x_{\lambda_{k-1}}^* - x_{\lambda_k}^*  \right)}_{Term4}.
    \end{aligned}$$ Applying the fact that $2a^Tb \leq \|a\|^2/\alpha + \alpha \|b\|^2 $ for all $a,b \in \mathbb{R}^n$ and $\alpha>0$ for $Term4$ when $\alpha = 2m/{\gamma_k \lambda_k \mu_h}$, we obtain $$\begin{aligned}
    &\left \|x_{k} - x_{\lambda_k}^* \right \|^2 =\left \|x_{k} - x_{\lambda_{k-1}}^* \right\|^2 + \left \|x_{\lambda_k}^* - x_{\lambda_{k-1}}^* \right\|^2 \\ & + \frac{\gamma_k \lambda_k \mu_h}{2m} \left \|x_{k} - x_{\lambda_{k-1}}^* \right\|^2 + \frac {2m} {\gamma_k \lambda_k \mu_h} \left \|x_{\lambda_k}^* - x_{\lambda_{k-1}}^* \right\|^2\\
    & = \left( 1+ \frac{\gamma_k \lambda_k \mu_h}{2m} \right) \left \|x_{k} - x_{\lambda_{k-1}}^* \right\|^2 + \left(1+ \frac{2m}{\gamma_k \lambda_k \mu_h}\right)\left \|x_{\lambda_k}^* - x_{\lambda_{k-1}}^* \right\|^2. 
    \end{aligned}$$ Using Lemma \[lemma:err\_inequalitygeneral\](a), we obtain $$\begin{aligned}
    &\left \|x_{k} - x_{\lambda_k}^* \right \|^2  \leq  \left( 1+ \frac{\gamma_k \lambda_k \mu_h}{2m} \right) \left \|x_{k} - x_{\lambda_{k-1}}^* \right\|^2 \\ 