Results 1 to 2 of 2

Thread: Is every strictly locally monotone sequence strictly monotone?

  1. #1
    Newbie
    Joined
    Sep 2007
    Posts
    5

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by outlandish View Post
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: May 15th 2011, 02:13 AM
  2. Monotone sequence
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Feb 8th 2011, 09:37 AM
  3. Set, monotone sequence, inf
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Nov 17th 2010, 12:52 PM
  4. monotone sequence
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Sep 26th 2009, 07:21 AM
  5. Show strictly monotone
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 11th 2007, 07:37 AM

Search Tags


/mathhelpforum @mathhelpforum