The sequence of vectors $(x_n)_n$ is bounded and such that $$ \left\| x_{n+1} - y\right\|^2 \leq \left\| x_n - y \right\|^2 - \alpha \left\| F_n(x_n) - x_n \right\|$$ for some $0 < a < 1$ and $y\in \text{fix}(F_n)$. $F_n : \mathbb{R}^m \rightarrow \mathbb{R}^m$ are $1$-Lipschitz continuous.
Define the set $S = \bigcap_{n =0}^{\infty} \text{fix}(F_n) \neq \emptyset$.
Does $\left\| x_n - y\right\|$ decreases monotonically until $x_n \in S$?
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What I have done:
- Shown that in the simpler case $F_k = F$ the claim is true by summing over the indices in both sides of the inequality.