# Is every strictly locally monotone sequence strictly monotone?

• Sep 26th 2007, 09:08 AM
outlandish
Is every strictly locally monotone sequence strictly monotone?
Sequence $a_n$, n=1,2,3... is strictly locally monotone if for every integer k>1 either $a_{k-1}$< $a_k$ < $a_{k+1}$ OR $a_{k-1}$> $a_k$> $a_{k+1}$.
Prove every strictly locally monotone sequence is strictly monotone or give counterexample.
• Sep 26th 2007, 10:01 AM
ThePerfectHacker
Quote:

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

We know for $k=2$ we have $a_1 < a_2 < a_3$ (without lose of generality). We also know for $k=3$ we have $a_2 < a_3 < a_4$ or $a_2 > a_3 >a_4$. It cannot be the latter for $a_1 < a_2 < a_3$. Thus, $a_2. So far we know, $a_1 < a_2 < a_3 < a_4$. Continue this argument.