mentioning that, in principle, set $\Omega$ can be empty and, in such case, $f^*=-\infty$. In the next lemma we analyze the behavior of the sequence $\{x_k\}$ under the hypothesis that $\Omega \neq \varnothing$.

\[Le:BoundExog\] If $\Omega \neq \varnothing$, then $\{x_k\}$ is quasi-Féjer convergent to $\Omega$. Consequently, $\{x_k\}$ is bounded.

Since $\Omega \neq \varnothing$, take $x\in \Omega$. Thus, by using the definition of $\Omega$ in and Lemma \[Le:FejerConvExog\], we conclude that $\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 + \rho \alpha_k^2$, for all $k=0, 1, \ldots$. Hence, using the first inequality in , the first statement of the lemma follows from Definition \[def:QuasiFejer\]. The second statement of the lemma follows from the first part of Theorem \[teo.qf\].

Now, we are ready to prove the main result of this section, which refers to the asymptotic convergence of $\{x_k\}$. We remark that in the first part of the next theorem we do not assume neither $\Omega^* \neq \varnothing$ nor that $f^*$ is finite.

\[teo.Main\] The following equality holds

$$\label{eq:linfs}
        \liminf_k f(x_k)=f^*.$$

In addition, if $\Omega^* \neq \varnothing$ then the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$.

Assume by contradiction that $\liminf_k f(x_k) > f^*$. In this case, we have $\Omega \neq \varnothing$. Consequently by Lemma \[Le:BoundExog\], we conclude 