such that $|x_j| \ge 1/\sqrt{n}$. Without loss of generality, let us suppose that this $x_j$ is positive.[^6]

The rest of the proof relies on a technique from [@LO81] to provide a lower bound on the right-hand side of Eq. (\[eq:vv\]). Let $$z_i = x_{i} - x_{i+1} \ \ \hbox{for } i<n,\quad
\hbox{and}\quad z_n=0.$$ Note that $z_i \geq 0$ for all $i$ Since $x_j\ge 1/\sqrt{n}$ for some $j$, Combining with [Eq. (\[eq:vv\])]{}, we obtain $$\frac{V(kB)-V((k+1)B)}{V(kB)} \geq
\frac{\eta}{2} \min_{z_i \geq 0,\ \sum_i z_i \geq 1/\sqrt{n}}
\sum_{i=1}^{n} z_i^2.$$ [The minimization problem on the right-hand side is a symmetric convex optimization problem, and therefore has a symmetric optimal solution, namely]{} $z_i = 1/n^{1.5}$ for all $i$. This results in an optimal value of $1/n^2$. Therefore, $$\frac{V(kB)-V((k+1)B)}{V(kB)} \geq \frac{\eta}{2 n^2},$$ which [is]{} the desired result.

We are now ready for our main result, which establishes that the convergence time of the sequence of vectors $x(k)$ generated by Eq.(\[noquant\]) is of order $O(n^2B/\eta)$.

\[uqbound\] Let Assumptions \[weights\] and \[weakconnect\] hold. Then, there exists an absolute constant[^7] $c$ such that we have $$V({k})
\leq \epsilon V(0)\qquad \hbox{for all } k\ge {c} (n^2/\eta)B \log
(1/\epsilon).$$

The result follows immediately from Lemma \[lboundvar\].

Recall that, according to Lemma \[as2implas3\], 