sequence $\{\z_n:n\in\N\}$ satisfies the finite length property.

Since the proof of \[Th1\] follows the framework given in [@C.Bao2016 Theorem 3.7.], we only need to mention the key component. Since $H(\z)\geq0$ satisfies the KL property, it can be easily verified that it satisfies the first condition in [@C.Bao2016 Theorem 3.7]. Interested readers may refer to [@C.Bao2016] for the details. Hence, we can complete the proof provided that we can verify that the sequence $\left\{\z_n:n\in\N\right\}$ generated by \[Alg1\] is bounded. In fact, since the constraints $\mC$ and $\mD$ are compact sets, it suffices to show that $\bc_n$ is bounded.

\[Lemma1\] Let $H(\z)$ be defined as in . For the sequence $\left\{\z_n:n\in\N\right\}$ generated by \[Alg1\], there exists a constant $C>0$ independent on $n$ such that $$\begin{aligned}
\left\|\bc_n\right\|_2\leq C~~~~~~~\text{for all}~~~n\in\N.\end{aligned}$$ Hence, $\bc_n$ is bounded.

Since $\bsv_n\in\mC$ for all $n\geq0$, we have $\left\|\bsv_n\right\|_2\leq RN$ for all $n\geq0$. Hence, the proof is completed by mathematical induction. For $n=0$, \[Initialize\] implies $$\begin{aligned}
\left\|\bc_0\right\|_2\leq\left\|\bsW_0\left(\La\bsv_0\right)\right\|_2=\left\|\La\bsv_0\right\|_2\leq\left\|\La\right\|\left\|\bsv_0\right\|_2\leq\left\|\La\right\|RN,\end{aligned}$$ where $\left\|\La\right\|$ denotes the spectral norm of $\La$, and the equality comes from the fact that $\bsW_0\in\mD$.

For the mathematical induction, we assume that $\left\|\bc_n\right\|_2\leq C:=\left\|\La\right\|RN$. Due to the hard thresholding in \[csubexplicit\], we have $$\begin{aligned}
\left\|\bc_{n+1}\right\|_2&\leq\left\|\f{\mu\bsW_{n}\left(\La\bsv_{n+1}\right)+\beta_2\bc_n}{\mu+\beta_2}\right\|_2\\
&\leq\f{\mu}{\mu+\beta_2}\left\|\bsW_{n}\left(\La\bsv_{n+1}\right)\right\|_2+\f{\beta_2}{\mu+\beta_2}\left\|\bc_n\right\|_2\\
&=\f{\mu}{\mu+\beta_2}\left\|\La\bsv_{n+1}\right\|_2+\f{\beta_2}{\mu+\beta_2}\left\|\bc_n\right\|_2\\
&\leq\f{\mu}{\mu+\beta_2}\left\|\La\right\|\left\|\bsv_{n+1}\right\|_2+\f{\beta_2}{\mu+\beta_2}\left\|\bc_n\right\|_2\leq\left\|\La\right\|RN=C,\end{aligned}$$ where the equality again 