IMC 2002 · Problem 6

For an $n \times n$ matrix $M$ with real entries let $\|M\| = \sup\limits_{x \in \mathbb{R}^n \setminus \{0\}} \dfrac{\|Mx\|_2}{\|x\|_2}$, where $\|\cdot\|_2$ denotes the Euclidean norm on $\mathbb{R}^n$. Assume that an $n \times n$ matrix $A$ with real entries satisfies $\|A^k - A^{k-1}\| \le \dfrac{1}{2002k}$ for all positive integers $k$. Prove that $\|A^k\| \le 2002$ for all positive integers $k$.