Results 1 to 2 of 2

Math Help - Measure theory

  1. #1
    Member SENTINEL4's Avatar
    Joined
    Mar 2009
    From
    Kastoria, Greece
    Posts
    92

    Measure theory

    Show that (a,b) = \bigcup_{n\in N}( a_{n},b_{n}) , where a,b\in\mathbb{R} , { a_{n}} is a decreasing sequence of rational numbers with a_{n}\rightarrow a , { b_{n}} is an increasing sequence of rational numbers with b_{n}\rightarrow b .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,412
    Thanks
    1328
    Suppose x is in (a, b). That is, that a< x< b. Show that, for some n, a_n< x< b_n. That should be easy. Take \epsilon=  x-a. Since a_n \to a, there exist n such that |a- a_n|< \delta so a<a_n< x. Since  b_n\to b, there exist m such that |b-b_n|< \delta so b< b_n< b.

    It should be clear that is x< a, then x is in NONE of (a_n, b_n) and so not in their union. Similarly for x> b. The important part is x= a and x= b. You need to show that a< a_n and b_n< b for all n.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Some measure Theory
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 24th 2010, 08:39 AM
  2. Measure theory
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 7th 2010, 10:55 AM
  3. Measure Theory
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 25th 2010, 07:20 AM
  4. Measure Theory
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: December 17th 2007, 09:29 PM
  5. measure theory
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: June 14th 2007, 01:47 AM

Search Tags


/mathhelpforum @mathhelpforum