Question 6
Let $x_1, x_2, \dots, x_n$ be positive real numbers. For each positive integer $k$, define
$$
S_k = x_1^k + x_2^k + \dots + x_n^k.
$$
(a) Prove that if $S_1 \lt S_2$ the sequence $S_1, S_2, S_3, \dots$ is strictly increasing.
(b) Prove that it is possible that $S_1 \gt S_2$ and yet the sequence $S_1, S_2, S_3, \dots$ is not strictly decreasing.
$$
S_k = x_1^k + x_2^k + \dots + x_n^k.
$$
(a) Prove that if $S_1 \lt S_2$ the sequence $S_1, S_2, S_3, \dots$ is strictly increasing.
(b) Prove that it is possible that $S_1 \gt S_2$ and yet the sequence $S_1, S_2, S_3, \dots$ is not strictly decreasing.