The goal is to obtain an upper bound for the norm of the vector $$ \left\|\sum\limits_{k=0}^{\infty}(I−A)^kAw_k\right\| $$ for any symmetric matrix $A\in{\mathbb R}^{n×n}$ which $0\preceq A\preceq I$ ($I$ is identity matrix) and for any vectors $w_k\in{\mathbb R}^n$ such that $\|w_k\|\leq1,\,\,k=0,1,\ldots$ (all norms are euclidean).

It is easy to show this norm is less $\sqrt{n}$, but it seems it should not depends on $n$.