Results 1 to 3 of 3

Math Help - Monotone Decreasing

  1. #1
    Member
    Joined
    Aug 2009
    Posts
    78

    Monotone Decreasing

    Let (f_n) be a non-negative sequence of monotone decreasing integrable functions on [0,1].
    Show that (f_n) \downarrow 0 if and only if (\int_{[0,1]} f_n) \downarrow 0.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by problem View Post
    Let (f_n) be a non-negative sequence of monotone decreasing integrable functions on [0,1].
    Show that (f_n) \downarrow 0 if and only if (\int_{[0,1]} f_n) \downarrow 0.
    Since the functions are non-increasing (monotome decreasing) it means that f_n(b) \leq f_n(x) \leq f_n(a) for each x\in [a,b]. Therefore, f_n(b) \leq \smallint_0^1 f_n \leq f_n(a). However, we are told that f_n(a),f_n(b)\to 0 therefore the integral, being squeezed between those two sequences, must go to zero too. Conversely it is not true. Just let f_n(a) = 1 with f_n(x) = 0 for a<x\leq b. Then each f_n(x) is integrable, non-negative, non-increasing with zero integral however f_n does not converge to the zero function.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2009
    Posts
    78
    the monotone decreasing that i refer to is that 0\le f_{n+1} \le f_n for all n.Is it still valid to use the argument above?
    Last edited by problem; September 8th 2009 at 07:17 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Monotone
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: December 29th 2011, 04:30 AM
  2. Replies: 3
    Last Post: May 15th 2011, 02:13 AM
  3. Set, monotone sequence, inf
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 17th 2010, 12:52 PM
  4. monotone
    Posted in the Calculus Forum
    Replies: 6
    Last Post: September 25th 2008, 10:17 PM
  5. Replies: 1
    Last Post: September 26th 2007, 10:01 AM

Search Tags


/mathhelpforum @mathhelpforum