Question: A sequence {$x_n$} is monotone. Prove it's arithmetic mean sequence, defined as {$y_n$}=($x_1$+$x_2$+...+$x_k$)/$k$, is also monotone.
My attempt: I first assumed $x_n$ is strictly increasing. I defined the mean sequence as $y_{k+1}$=($k$$y_n$+$x_{n+1}$)/$k+1$.
Then $y_{k+1}$ $\bullet$ ($k+1$) = $k$$y_n$+$x_{n+1}$ which implies $k$($y_{k+1}$-$y_k$)=$x_{k+1}$ - $y_{k+1}$.
Doing a few problems, I can see that $x_{k+1}$ - $y_{k+1}$ > 0 but I am struggling to show this using the definition of arithmetic mean given (it can written with summation notation, but I am new to typesetting.)
I know a monotone sequence is also Archimedean. I've tried using induction (unsuccessfully).
Initial question: Monotone sequence property says that a sequence may be strictly increasing/decreasing or non-increasing/non-decreasing. May I just show 2 cases, strictly increasing and strictly decreasing?