Question 80
Let $a_1, a_2, \dots, a_n, \dots$ be a sequence of positive real numbers. For every positive integer $n$, write
$$
s_n = a_1 + a_2 + \dots + a_n \quad \text{and} \quad \sigma_n = \frac{a_1}{1+a_1} + \frac{a_2}{1+a_2} + \dots + \frac{a_n}{1+a_n}.
$$
Prove that, if the $s_n$ form an unbounded sequence, then so do the $\sigma_n$.
$$
s_n = a_1 + a_2 + \dots + a_n \quad \text{and} \quad \sigma_n = \frac{a_1}{1+a_1} + \frac{a_2}{1+a_2} + \dots + \frac{a_n}{1+a_n}.
$$
Prove that, if the $s_n$ form an unbounded sequence, then so do the $\sigma_n$.