Is every strictly locally monotone sequence strictly monotone?

• Sep 26th 2007, 09:08 AM
outlandish
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.
• Sep 26th 2007, 10:01 AM
ThePerfectHacker
Quote:

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.