Prove that a sequence is nondecreasing sequence. **The question is below.
Let $(s_n)$ be a nondecreasing sequence of positive numbers and define $\sigma_n=\frac{1}{n}(s_1+s_2+\cdots+s_n)$. Prove that $(\sigma_n)$ is a nondecreasing sequence.
**In my text book, the monotone sequence is defined as below.
A sequence$(s_n)$ of real numbers is called a nondecreasing sequence if $s_n \leq s_{n+1}, \forall n$, and nonincreasing sequence if $s_n \geq s_{n+1}, \forall n$.
**My work is as follows.
$$\sigma_1 = \frac{s_1}{1} \Rightarrow \sigma_1 = s_1$$
$$\sigma_2 = \frac{s_1+s_2}{2}\Rightarrow 2*\sigma_2 = s_1+s_2$$
$$\sigma_3 = \frac{s_1+s_2+s_3}{3}\Rightarrow 3*\sigma_3 = s_1+s_2+s_3$$
$$\vdots$$
$$\sigma_n = \frac{s_1+s_2+\cdots+s_n}{n}\Rightarrow n*\sigma_n = s_1+s_2+\cdots+s_n$$
Since $(s_n)$ is nondecreasing sequence of positive numbers and $1*\sigma_1 \leq 2*\sigma_2 \leq \cdots \leq n*\sigma_n$, $\sigma_n$ is nondecreasing sequence. Q.E.D.
I am afraid is could be wrong because I proved it with coefficient for each sequence $\sigma_n$. Can anyone give me some hints if this is wrong??
 A: Suppose that $$\frac1{n+1}(s_1+\ldots+s_n+s_{n+1})<\frac1n(s_1+\ldots+s_n)\;;$$ then 
$$\begin{align*}s_1+\ldots+s_n+s_{n+1}&<\frac{n+1}n(s_1+\ldots+s_n)\\
&=\left(1+\frac1n\right)(s_1+\ldots+s_n)\\
&=(s_1+\ldots+s_n)+\frac1n(s_1+\ldots+s_n)\;,
\end{align*}$$
and therefore $$s_{n+1}<\frac1n(s_1+\ldots+s_n)\;.$$ But then $$ns_{n+1}<s_1+\ldots+s_n\le ns_n\;,$$ and $s_{n+1}<s_n$, which is impossible.
A: I don't quite follow your proof; there are decreasing sequences $\{a_k\}$ satisfying $k a_k \le (k + 1) a_{k + 1}$ for every $k$.
As an easier solution, just consider
\begin{align}
\sigma_{n + 1} - \sigma_n &= \frac{s_1 + s_2 + ... + s_n + s_{n + 1}}{n + 1} - \frac{s_1 + s_2 + ... + s_n}{n} \\
&= \frac{1}{n(n + 1)} \left(n s_1 + n s_2 + ... + n s_n + n s_{n + 1} - (n + 1) s_1 - ... - (n + 1) s_n\right) \\
&= \frac{1}{n(n + 1)} \left(n s_{n + 1} - s_1 - s_2 - ... - s_n\right) \\
&= \frac{1}{n(n + 1)} \left((s_{n + 1} - s_1) + (s_{n + 1} - s_2) + ... + (s_{n + 1} - s_n)\right) \\
&\ge 0
\end{align}
Because every term is non-negative, since $\{s_n\}$ is monotone increasing.
