2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2)\\
    & + 2\gamma_k \sum_{i=0}^{m-1} \left( \underbrace{|f_{i+1}(x_{k,i}) - f_{i+1}(x_{k}) |}_{Term2} + \frac{\lambda_k}{m} \underbrace{|h(x_{k,i})  -h(x_{k})|}_{Term3}  \right),
    \end{aligned}$$]{}]{} where $Term1\leq 0$ is used due to optimality of $x_{\lambda_k}^*$ for $f+\lambda_k h$. Also, from Remark \[bound on gradient by beck\](b) we know that $Term2 \leq C_f\|x_{k,i} - x_k\|$ and $Term3 \leq C_h\|x_{k,i} - x_k\|$. So, We have $$\begin{aligned}
\label{incrementalineq1}
    &\left \|x_{k+1} - x_{\lambda_k}^* \right \|^2  \leq \left( 1- \frac{\gamma_k \lambda_k \mu_h}{m} \right) \left \|x_{k} - x_{\lambda_k}^* \right \|^2  \nonumber \\ &+ 2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2)
    + 2 (C_f+\lambda_k C_h) \gamma_k \sum_{i=0}^{m-1} \| x_{k,i} - x_k\|.
    \end{aligned}$$ Next, we find an upper bound for $\| x_{k,i} - x_k\|$. We have\
$    \| x_{k,1} - x_k\|\\ =\left \|\mathcal{P}_X \left(x_{k,0} - \gamma_k\left(g_{f_1}(x_{k,0}) + \frac{\lambda_k}{m} g_h(x_{k,0})\right) \right)- \mathcal{P}_X(x_k) \right \|\\
    \leq \gamma_k \left \|g_{f_1}(x_{k,0}) + \frac{\lambda_k}{m} g_h(x_{k,0})\right \| \leq \gamma_k \left( C_f + \frac{\lambda_k}{m} C_h \right).\\
    $ For $i>0$, in a similar way, we have $$\begin{aligned}
    \| x_{k,i+1} - x_k\|         \leq \| x_{k,i} - x_k\|+ \gamma_k \left( C_f + \frac{\lambda_k}{m} C_h \right).
    \end{aligned}$$ So for $i=0,1, \cdots, m-1$, we have $$\begin{aligned}
 \label{ineq2 INC}
    \| x_{k,i+1} - x_k\| &\leq (i+1)\gamma_k \left( C_f + \frac{\lambda_k}{m} C_h \right) \nonumber \\&\leq  (i+1)\gamma_k 