Results 1 to 3 of 3

Math Help - convergent sequence

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    172

    convergent sequence

    prove \ that \ if \ a_n \ is \ an \ increasing \ sequence \ , \ and \ b_n \ is \ a \ decreasing  \ sequence \ with \ a_n \ \leq \ b_n \ , \ \forall n\in N \ then \ both
    a_n \ and \ b_n \ are \ convergent \ sequences
     \ \ with \ lim \ a_n \leq lim \ b_n
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,607
    Thanks
    1574
    Awards
    1
    Do you know that any bounded monotone sequence converges?
    Do you understand how b_1 is an upper bound for the (a_n) sequence?
    What is a lower bound for the (b_n) sequence?
    Now put this together.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Abu-Khalil's Avatar
    Joined
    Oct 2008
    From
    Santiago
    Posts
    148
    Suppose a_n\to a,b_n\to b and a>b. Hence, \exists N_1:a-a_n>a-\frac{a+b}{2},\forall n>N_1 and \exists N_2:b_n-b>\frac{a+b}{2}-b,\forall n>N_2. So for any n\geq \max\left\{N_1,N_2\right\} you have a_n>b_n.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Convergent sequence
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: October 15th 2010, 05:27 AM
  2. convergent sequence
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: February 27th 2010, 11:59 AM
  3. Convergent Sequence
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: August 11th 2009, 07:42 AM
  4. Convergent sequence
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: July 7th 2009, 04:11 PM
  5. convergent sequence
    Posted in the Calculus Forum
    Replies: 8
    Last Post: October 4th 2007, 12:37 PM

Search Tags


/mathhelpforum @mathhelpforum