# Thread: Is every strictly locally monotone sequence strictly monotone?

1. ## Is every strictly locally monotone sequence strictly monotone?

Sequence $\displaystyle a_n$, n=1,2,3... is strictly locally monotone if for every integer k>1 either $\displaystyle a_{k-1}$<$\displaystyle a_k$ < $\displaystyle a_{k+1}$ OR $\displaystyle a_{k-1}$> $\displaystyle a_k$> $\displaystyle a_{k+1}$.
Prove every strictly locally monotone sequence is strictly monotone or give counterexample.

2. Originally Posted by outlandish
Sequence $\displaystyle a_n$, n=1,2,3... is strictly locally monotone if for every integer k>1 either $\displaystyle a_{k-1}$<$\displaystyle a_k$ < $\displaystyle a_{k+1}$ OR $\displaystyle a_{k-1}$> $\displaystyle a_k$> $\displaystyle a_{k+1}$.
Prove every strictly locally monotone sequence is strictly monotone or give counterexample.
We know for $\displaystyle k=2$ we have $\displaystyle a_1 < a_2 < a_3$ (without lose of generality). We also know for $\displaystyle k=3$ we have $\displaystyle a_2 < a_3 < a_4$ or $\displaystyle a_2 > a_3 >a_4$. It cannot be the latter for $\displaystyle a_1 < a_2 < a_3$. Thus, $\displaystyle a_2<a_3<a_4$. So far we know, $\displaystyle a_1 < a_2 < a_3 < a_4$. Continue this argument.