In [1], there are many theorems and propositions on (quasi) Fejér monotonicity.
Let us focus on finite-dimensional spaces, for instance Euclidean space.
Theorem 3.3 and Proposition 4.1-4.3 in [1] (modified for finite dimensional and adapted to quasi-Fejér): Let $C$ be a nonempty closed subset of the Euclidean space $\mathbb{R}^n$ and $(x_k)_{k\geq0}$ be a sequence in $\mathbb{R}^n$. Additionally, let $\{W_k\}$ be a positive definite matrix such that $W_{k+1} \preceq \left(1+\alpha_k\right)W_{k}$, where $\sum_{k=0}^\infty \alpha_k < \infty$. Assume that,
- for all $z \in C$ and $k \geq 0$, $\| x_{k+1} - z \|^2_{W_{k+1}} \leq \left(1+\alpha_k\right)\| x_{k} - z \|^2_{W_{k}} + \epsilon_k$, where $\sum_{k=0}^\infty \epsilon_k < \infty$,
- every sequential cluster point of $(x_k)_{k\geq0}$ belongs to $C$.
Then, $(x_k)_{k\geq0}$ converges (strongly?) to an element in $C$.
Question 1: Do I understand the reference [1] correctly?
Question 2: Assuming yes to my question 1, if we replace one of the above condition $W_{k+1} \preceq \left(1+\alpha_k\right)W_{k}$ with $W_{k+1} \preceq W_{k} + P_k$, where $P_k \succeq 0$ is a positive semidefinite diagonal matrix and $\sum_{k=0}^\infty P_k[i,i] < \infty \ \forall i$ (all the diagonal elements are summable), can the above theorem still hold true? If yes, can we prove it?
The highlighted statement is not true in general. E.g., suppose that $W_k=\dfrac1{k^2}\,I$, where $I$ is the identity matrix, $\alpha_k=0$, $\epsilon_k=1/k^2$, $C$ is any closed set of diameter $1$, $u$ and $v$ are any two distinct points in $C$, and $x_k$ alternates between $u$ and $v$.
Then all the conditions of the highlighted statement are satisfied, but the sequence $(x_k)$ does not converge.