that $\{x_k\}$ is bounded. Letting $x\in \Omega$, there exist $\tau>0$ and $k_0\in\mathbb{N}$ such that $f(x)<f(x_k)-\tau,$ for all $k\geq k_0$. Hence, using Lemma \[Le:FejerConvExog\], we have $$\label{eq:bXk}
        \|x_{k+1}-x\|^2 \leq \|x_{k}-x\|^2 + \rho \alpha_k^2- 2 \frac{\alpha_ k}{\eta_k} \tau, \qquad  k=k_0, k_0+1, \dots.$$ On the other hand, it follows from that the sequence $\left\lbrace \epsilon_k \right\rbrace$ is bounded. Thus, considering that $\{x_k\}$ is bounded, Proposition \[pr:CompE-subdif\] implies that $\{s_k\}$ is also bounded. Let $c > 0$ be such that $\|s_k\| \leq c$, for all $k \geq  0$. Hence, using second equality in , we have $\eta_k = \max\left\{1, \| s_k\|\right\} \leq \max\left\{1, c \right\} =: \varGamma$. Thus, letting $\ell \in \mathbb{N}$ and using , we conclude that $$\frac{2\tau}{\varGamma}\sum_{j=k_0}^{\ell+k_0}\alpha_j \leq \|x_{k_0}-x\|^2 - \|x_{k_0+ \ell+1}-x\| + \rho \sum_{j=k_0}^{\ell+k_0}\alpha^2_j \leq \|x_{k_0}-x\|^2 + \rho \sum_{j=k_0}^{\ell+k_0}\alpha^2_j.$$ Since the last inequality holds for all $\ell \in \mathbb{N}$ then, by using the first two conditions on $\{\alpha_k\}$ in , we have a contraction. Therefore, holds. For proving the last statement, let us assume that $\Omega^*\neq\varnothing$. In this case, we also have $\Omega\neq\varnothing$ and, from Lemma \[Le:BoundExog\], the sequence $\{x_k\}$ is bounded and quasi-Féjer convergent to $\Omega$. The equality implies that $\{f(x_k)\}$ has a decreasing monotonous subsequence $\{f(x_{k_j})\}$ such that $\lim_{j\rightarrow 