convergence for symmetric, positive semi-definite operator

Question

Assume $u$ is a vector in the Euclidean space $\mathbb{R}^N$, $||u||=\sqrt{\langle u, u\rangle}$, where $\langle u, v\rangle = \sum_{i=1}^N u_i v_i$. I have that $||u^{k+1}-u||\leq ||I - c A||||u^k-u||$, where $A$ is symmetric, positive semi-definite operator and $c>0$ is a parameter. Here $||(I-cA)||$ is the operator norm of $(I-cA)$. I know that $||I - c A|| = max(|1-c \lambda_{min}|,|1-c\lambda_{max}|)$, where $\lambda_{min}$ and $\lambda_{max}$ is the least and the greatest eigenvalue of $A$, respectively. Therefore, we have $||I - c A|| = max(1,|1-c\lambda_{max}|)$ and for $c<2/\lambda_{max}$ we get $||u^{k+1}-u||\leq||u^k-u||$. How to get that $||u^{k+1}-u||< ||u^k-u||$? Or in general, to show that $||u^k-u||\rightarrow 0$ as $k\rightarrow\infty$?