Consider the vector $$x_k=\sum\limits_{i=0}^k(I-A)^i A w_{k-i}$$ where symmetric matrix $A\in\mathbb R^{n\times n}$ such that $0\preceq A\preceq I$ (in positive-definiteness sense, that is $0\le\lambda(A)\le1$) and arbitrary vectors $w_k\in\mathbb R^n$ such that $\|w_k\|\le1$.
The goal is to obtain an upper bound for $$\|x_k\| \quad\text{when}\quad k\to\infty$$
All norms are Euclidean.
The estimate should not include matrix $A$ itself, because we know only its 'bounds'.