$\hat u\in\mathbb{R}^n$ and let $\{\alpha_k\}$ be a sequence of positive numbers where $\sum_{k=0}^\infty \alpha_k=\infty$. Suppose $\{v_k\}$ is given by $v_k\triangleq \left({\sum_{t=0}^{k-1} \alpha_t u_t}\right)/{\sum_{t=0}^{k-1} \alpha_t}$ for all $k\ge1$. Then, $\lim_{k \rightarrow \infty} v_k=\hat u.$

Now, we can illustrate our ultimate goal in this section which is showing the convergence of the sequence $\{\bar{x_k}\}$ generated by Algorithm \[algorithm:IRIG\] to $x_h^*$

\[thm conv for xbar INC\] Consider problem . Let Assumption \[assum:properties\] hold. Also 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 $\gamma_0\lambda_0 \leq \frac{2m}{\mu_h}$. Let $\{\bar x_k\}$ be generated by Algorithm \[algorithm:IRIG\]. If $a>b$, $a>0.5$, $a+b<1$ and $ar \leq 1$, then $\{\bar x_k\}$ converges to $x_h^*$.

Considering the given assumptions, by Lemma \[lemma condition for sequences INC\] we can see that Assumption \[assumptions for conv INC\] holds. We have $$\begin{aligned}
    \|\bar x_{k+1}-x_h^*\|=\left\|\sum_{t=0}^{k} \psi_{t,k} x_t-\sum_{t=0}^{k}\psi_{t,k} x_h^*\right\|\leq \sum_{t=0}^{k} \psi_{t,k}\|x_t-x_h^*\|,
        \end{aligned}$$ applying $\sum_{t=0}^{k} \psi_{t,k}=1$ from and the triangular inequality. Now consider the definition of $\psi_k$ and let $\alpha_t \triangleq \gamma_t^r$, $u_t \triangleq \|x_t-x_h^*\|$ and $v_{k+1} \triangleq \sum_{t=0}^{k} \psi_{t,k} \|x_t-x_h^*\|$. Since $ar \leq1$ we have $\sum_{t=0}^\infty \alpha_t=\sum_{t=0}^\infty \gamma_t^{r}=\sum_{t=0}^\infty (t+1)^{-ar}=\infty$. The sequence $\{\lambda_t \}$ is decreasing to zero due to $b>0$, So, from Proposition \[conv x\_k 