Results 1 to 2 of 2

Math Help - Real Analysis - Sequences #3

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    135

    Real Analysis - Sequences #3

    This question has three individual questions to it, so I'll post them as b,c and d. The instructions say to decide which of the following conjectures are true and which are false. Supply a proof for those that are valid and a counterexample for each that is not valid. Here they are:

    b. If fn --> f uniformly on A and g is a bounded function on A, then fn*g --> fg uniformly on A.

    c. If fn --> f uniformly on a set A, and if each fn is bounded on A, then f must also be bounded.

    d. If fn --> f uniformly on a set A, and if fn --> f uniformly on a set B, then fn --> f uniformly on A U B.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by ajj86 View Post
    This question has three individual questions to it, so I'll post them as b,c and d. The instructions say to decide which of the following conjectures are true and which are false. Supply a proof for those that are valid and a counterexample for each that is not valid. Here they are:

    b. If fn --> f uniformly on A and g is a bounded function on A, then fn*g --> fg uniformly on A.

    c. If fn --> f uniformly on a set A, and if each fn is bounded on A, then f must also be bounded.

    d. If fn --> f uniformly on a set A, and if fn --> f uniformly on a set B, then fn --> f uniformly on A U B.
    Note: You may want to wait for confirmation on my solutions/suggestions by more seniro members

    Which are you having trouble with? I dont have super much time but Ill try one'

    b. True, Im assuming that f_n is totally bounded. So let |f_n(x)|\leqslant M, so choose N such that N>n \implies |f_n(x)-f(x)|<1 or f_n(x)-1<f(x)<1+f_n(x) this in turn implies -(M+1)<f(x)<M+1
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Real Analysis - Sequences 3
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: February 16th 2010, 03:18 PM
  2. Real Analysis - Sequences 2
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 15th 2010, 07:43 PM
  3. Real Analysis - Sequences
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 15th 2010, 07:36 PM
  4. Real Analysis - Sequences #2
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 20th 2009, 03:57 PM
  5. Real Analysis - Sequences
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 20th 2009, 03:04 PM

Search Tags


/mathhelpforum @mathhelpforum