IMC 2002 Problem 6

For an $n\times n$ matrix with real entries let $||M||=\sup_{x\in \mathbb{R}^{n}\setminus\{0\}}\frac{||Mx||_{2}}{||x||_{2}}$ , where $||\cdot||_{2}$ denotes the Euclidean norm on $\mathbb{R}^{n}$ . Assume that an $n\times n$ matrxi $A$ with real entries satisfies $||A^{k}-A^{k-1}||\leq\frac{1}{2002k}$ for all positive integers $k$ . Prove that $||A^{k}||\leq 2002$ for all positive integers $k$ .