. Let Assumption \[assum:properties\] and \[assumptions for conv INC\] hold and $\{x_k\}$ be generated by Algorithm \[algorithm:IRIG\]. Then,

-   $\lim_{k \rightarrow \infty} \|x_k-x_{\lambda_{k-1}}^*\|^2=0$.

-   If $\lim_{k \rightarrow \infty} \lambda_k=0$, $x_k$ converges to the unique optimal solution of problem , i.e., $x_h^*$.

\(a) Consider the result from Lemma \[lemma: an error bound INC\]. We let $$\begin{aligned}
    \nu_k &\triangleq \|x_k - x_{\lambda_{k-1}}^*\|^2, \ \alpha_k \triangleq \frac{\gamma_k\lambda_k \mu_h}{2m}\\ \beta_k &\triangleq \frac{3m C_h^2}{\gamma_k \lambda_k \mu_h^3}\left(1-\frac{\lambda_{k-1}}{\lambda_k} \right)^2 +  6 m^2 \gamma_k^2 (C_f^2 + \lambda_k^2 C_h^2).
    \end{aligned}$$ From Assumption \[assumptions for conv INC\](a,b,c), since $\{\gamma_k\}$ and $\{\lambda_k\} $ are positive and $\gamma_0\lambda_0 \leq 2m/\mu_h$, we have $0 \leq \alpha_k \leq 1$ and $ \beta_k \geq0$ and also $\sum_{k=1}^{\infty}\alpha_k=\infty$ and $\sum_{k=1}^{\infty}\beta_k <\infty$. To show that all the necessary assumptions for Lemma \[lemma conv lemma\] are satisfied, we have $$\begin{aligned}
    \lim_{k \to \infty} \frac{\beta_k}{\alpha_k}    &=\frac{6m^2 C_h^2}{\mu_h^4}\lim_{k \to \infty} \frac{1}{\gamma_k^2\lambda_k^2}\left(\frac{\lambda_{k-1}}{\lambda_k}-1\right)^2 +\frac{12m^3C_f^2}{\mu_h}\lim_{k\to \infty} \frac{\gamma_k}{\lambda_k} \\& +
    \frac{12m^3C_h^2}{\mu_h}\lim_{k\to \infty} \gamma_k\lambda_k.
    \end{aligned}$$ Considering Assumption \[assumptions for conv INC\](e,f), we only need to show that $\lim_{k\to \infty} \gamma_k\lambda_k=0$. Since $\{\lambda_k\}$ is non-increasing for all $k\geq 0$, we have $\lambda_0^2 {\gamma_k}/{\lambda_k} \geq \gamma_k\lambda_k$. So by Assumption \[assumptions for conv INC\](f), $\lim_{k \to \infty} \gamma_k\lambda_k=0$. Consequently $\lim_{k \to \infty} \frac{\beta_k}{\alpha_k}=0$. Now Lemma \[lemma conv lemma\] 