IMC 2000 · Problem 4

a) Show that if $(x_i)$ is a decreasing sequence of positive numbers then

$$\left( \sum_{i=1}^{n} x_i^2 \right)^{1/2} \le \sum_{i=1}^{n} \frac{x_i}{\sqrt{i}}.$$

b) Show that there is a constant $C$ so that if $(x_i)$ is a decreasing sequence of positive numbers then

$$\sum_{m=1}^{\infty} \frac{1}{\sqrt{m}} \left( \sum_{i=m}^{\infty} x_i^2 \right)^{1/2} \le C \sum_{i=1}^{\infty} x_i.$$