Is every strictly locally monotone sequence strictly monotone?

**outlandish** · September 26th 2007, 09:08 AM

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.

**ThePerfectHacker** · September 26th 2007, 10:01 AM

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, $a_2<a_3<a_4$ . So far we know, $a_1 < a_2 < a_3 < a_4$ . Continue this argument.

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