Prove that, for positive real numbers $a$ and $b$,

$$2 \sqrt{a b} \leqslant a+b$$

For positive real numbers $a_{1}, a_{2}, \ldots$, prove that the convergence of

$$\sum_{n=1}^{\infty} a_{n}$$

implies the convergence of

$$\sum_{n=1}^{\infty} \frac{\sqrt{a_{n}}}{n}$$