For all $n \in \mathbb{N}$, let $$ x_{n+1} = x_n + \epsilon_n \left( f_n( x_n ) - x_n \right) $$
where $0 < \underline{\epsilon} \leq \epsilon_n \leq 1$, $\ f_n: \mathbb{R}^m \rightarrow C$ is $1$-Lipschitz continuous for all $n$, $C \subset \mathbb{R}^m$ is compact, and $x_0 \in C$.
Suppose that $F = \bigcap_{n \in \mathbb{N}} \text{fix}(f_n) \neq \emptyset$, and that for all $n$ and $y \in F$: $$ \left\| x_{n+1}-y\right\|^2 \leq \left\| x_{n}-y\right\|^2 - \rho \left\| x_{n} - f_n(x_n)\right\|^2$$ for some $0 < \rho < 1$.
Is it true that all the cluster points of $\left( x_n \right)_n$ are in $F$?
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What I have done:
0) Observed that, for all $y \in F$, the sequence $\left( \left\| x_n - y\right\| \right)_n$ is bounded and monotonically non-increasing, so it convergences.
1) Shown that $\lim_{n \rightarrow \infty} \left\| f_n(x_n) - x_n \right\| = 0$.
2) Shown that $\lim_{n \rightarrow \infty} \left\| x_{n+1} - x_n \right\| = 0$.