- f(x_k)\right) \nonumber\\
    & + 2 m \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2)   + 2 m^2 C_f \gamma_k^2 \left( C_f + \lambda_k C_h \right) + 4\gamma_k \lambda_k M_h.\label{ineq3 INC} 
    \end{aligned}$$ Multiplying both sides by $\gamma_k^{r-1}$ and adding and subtracting $\gamma_{k-1}^{r-1} \|x_k - x^*\|^2$ to the left hand side, we have $$\begin{aligned}
    &\gamma_k^{r-1} \left \|x_{k+1} - x^* \right \|^2  - \gamma_{k-1}^{r-1} \|x_k - x^*\|^2  + \left( \gamma_{k-1}^{r-1}-\gamma_k^{r-1}\right)  \left \|x_k - x^* \right \|^2 \\& \leq 2 \gamma_k^r \left( f^* - f(x_k)\right) + 2 m \gamma_k^{r+1} (C_f^2 + \lambda_k^2 C_h^2) \\
    & + 2 m^2 C_f \gamma_k^{r+1} \left( C_f + \lambda_k C_h \right) + 4 \gamma_k^r \lambda_k M_h.
    \end{aligned}$$ Since $\{\gamma_k\}$ in non-increasing and $r<1$ we have $ \gamma_{k-1}^{r-1} \leq \gamma_k^{r-1} $. Also, by the triangle inequality $\left \|x_k - x^* \right \|^2 \leq 2\|x_k\|^2+2\|x^*\|^2 \leq 4M^2$. So, we obtain $$\begin{aligned}
    &\gamma_k^{r-1} \left \|x_{k+1} - x^* \right \|^2  - \gamma_{k-1}^{r-1} \|x_k - x^*\|^2  + 4M^2\left( \gamma_{k-1}^{r-1}-\gamma_k^{r-1}\right)\\ & \leq 2 \gamma_k^r \left( f^* - f(x_k)\right) + 2 m \gamma_k^{r+1} (C_f^2 + \lambda_k^2 C_h^2) \\
    & + 2 m^2 C_f \gamma_k^{r+1} \left( C_f + \lambda_k C_h \right) + 4 \gamma_k^r \lambda_k M_h.
    \end{aligned}$$ Taking summation over $k=1,2, \cdots, N-1$, we obtain $$\begin{aligned}
    &\gamma_{N-1}^{r-1} \left \|x_N 