# Properties of the sequence

Given a sequence $$\{a_i\}_{i\in \mathbb{Z}},$$ consider the sequence defined by $$b_i:=F(a_{i-1},a_{i},a_{i+1}),$$ where $$F:\mathbb{R}^3 \rightarrow \mathbb{R}$$ is increasing in each of the variable, and $$F(a,a,a)=a.$$ Suppose total variation of the sequence $$\{a_i\}_{i\in \mathbb{Z}}$$ is finite i.e. $$\sup\limits_{k\in \mathbb{N}} \sum_{i=-k}^k |a_i-a_{i-1}| < \infty$$ then how to prove the following

$$\sup\limits_{k\in \mathbb{N}} \sum_{i=-k}^k |b_i-b_{i-1}| \leq \sup\limits_{k\in \mathbb{N}} \sum_{i=-k}^k |a_i-a_{i-1}|$$

I have an intuitive idea of the proof but some how unable to give a rigorous proof.

The idea is local extremas of $$b_i$$ are bounded below and above by some $$a_m$$ and $$a_n$$ because of the monotonicity..How to give a rigorous proof?

I could not succeed with mathematical induction either.. :(

Define $$F(x,y,z)=max(x,z),$$ clearly $$F$$ is increasing in each of the variable.
define the sequence by $$$$a_i= { \left\{ \begin{array}{ccl} 1 & \, \mbox{if}\,i=0, \\ 0 & \, \mbox{otherwise}\,, \end{array}\right.}$$$$ Then $$b_i$$ turns out to be
$$$$b_i= { \left\{ \begin{array}{ccl} 1 & \, \mbox{if}\,i=-1,1, \\ 0 & \, \mbox{otherwise}\,, \end{array}\right.}$$$$ and we have, $$\sup\limits_{k\in \mathbb{N}} \sum_{i=-k}^k |a_i-a_{i-1}|=2$$
$$\sup\limits_{k\in \mathbb{N}} \sum_{i=-k}^k |b_i-b_{i-1}| =4$$