I am trying to understand the argument for convergence here (page 16) for damped iterations of non-expansive maps.
Say we have a sequence $\{x^n\}_{n=0}^{\infty}$ that is generated as $x^n = \theta F(x^{n-1})+(1-\theta)x^{n-1}$ for a given map $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ which is non-expansive, i.e.$\|f(x) - f(y)\| \leq \|x-y\|$ for all $x,y\in\mathbb{R}^n$ and $\theta \in (0,1)$.
It follows (using an identity) that $$\sum_{j=0}^{k}\|F(x^j) - x^j\|_2^2 \leq \frac{\|x^0-x^\star\|_2^2}{\theta(1-\theta)}$$ and so clearly $\|F(x^n) - x^n\|_{2}^{2} \rightarrow 0$. However, I fail to see how this implies $$ \min_{j=0,1,...,k}\|F(x^j)-x^j\|_2^2 \leq \frac{\|x^0 - x^\star\|_2^2}{(k+1)\theta(1-\theta)}$$ and even if it did, how does this imply convergence of $\{x^k\}$ to a fixed-point?