can be applied. We have $$\begin{aligned}
    \lim_{k \to \infty} \nu_k = \lim_{k \to \infty} \|x_k - x_{\lambda_{k-1}}^*\|^2=0.
    \end{aligned}$$

\(b) Applying the triangular inequality, we obtain $$\begin{aligned}
    \|x_k -x_h^*\|^2 \leq 2\|x_k - x_{\lambda_{k-1}}^* \|^2 + 2 \|x_{\lambda_{k-1}}^*-x_h^*\|^2, \ \hbox{for all } k \geq 0.
    \end{aligned}$$ From part (a), $\|x_k -x_h^*\|^2$ converges to zero. Also, from Lemma \[lemma:err\_inequalitygeneral\](b), we know that when $\lambda_k \rightarrow 0$ the sequence $\{x_{\lambda_k}^*\}$ converges to the unique optimal solution of problem , i.e., $x_h^*$. Therefore the result holds.

To have previous proposition work, we require that sequences $\{\gamma_k\}$ and $\{\lambda_k\}$ satisfy Assumption \[assumptions for conv INC\]. Below, we provide a set of feasible sequences for this assumption. The proof is analogous to that of Lemma 5 in [@Yousefian17].

\[lemma condition for sequences INC\] Assume $\{\gamma_k\}$ and $\{\lambda_k\}$ are sequences such that $\gamma_k=\frac{\gamma_0}{(k+1)^a}$ and $\lambda_k=\frac{\lambda_0}{(k+1)^b}$ where $a,b,\gamma_0$ and $\lambda_0$ are positive scalars and $\gamma_0\lambda_0 \leq \frac{2m}{\mu_h}$. If $a>b$, $a>0.5$ and $a+b<1$, then the sequences $\{\gamma_k\}$ and $\{\lambda_k\}$ satisfy Assumption \[assumptions for conv INC\].

The following is a useful lemma in proving convergence that we will apply in Theorem \[thm conv for xbar INC\].

\[lemma thm for avr\] Let $\{u_t\}\subset \mathbb{R}^n$ be a convergent sequence with the limit point 