TSTST 2023 Problem 2

Let $n \geq m \geq 1$ be integers. Prove that $$\sum_{k=m}^{n}\left(\frac{1}{k^{2}}+\frac{1}{k^{3}}\right) \geq m \cdot\left(\sum_{k=m}^{n} \frac{1}{k^{2}}\right)^{2}$$