Results 1 to 2 of 2

Thread: Real Analysis - Sequences #2

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    135

    Real Analysis - Sequences #2

    Hey, I have been having trouble ever since I took part 1 of this class. So, here is question 2:

    Assume that (fn) converges uniformly to f on A and that each fn is uniformly continuous on A. Prove that f is uniformly continuous on A.

    Any help would be greatly appreciated. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by ajj86 View Post
    Hey, I have been having trouble ever since I took part 1 of this class. So, here is question 2:

    Assume that (fn) converges uniformly to f on A and that each fn is uniformly continuous on A. Prove that f is uniformly continuous on A.

    Any help would be greatly appreciated. Thanks.
    Here is the idea of the proof.

    Since $\displaystyle f_n \to f$ uniformly there exits an $\displaystyle N \in \mathbb{N}$ such that for $\displaystyle n > N$$\displaystyle |f_n-f|< \frac{\epsilon}{3}$ Now fix $\displaystyle M > N$

    Since each $\displaystyle f_n$ is uniformly continous there exists a delta such that $\displaystyle |x-y|< \delta \implies |f_M(x)-f_M(y)|< \frac{\epsilon}{3}$

    Now if $\displaystyle |x-y|<\delta$ then

    $\displaystyle |f(x)-f(y)|= |f(x)-f_M(x)+f_M(x)-f_M(y)+f_M(y)-f(y)|$

    $\displaystyle <|f(x)-f_M(x)|+|f_M(x)-f_M(y)|+|f_M(y)-f(y)|< \frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsil on}{3}=\epsilon $
    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: Feb 16th 2010, 02:18 PM
  2. Real Analysis - Sequences 2
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Feb 15th 2010, 06:43 PM
  3. Real Analysis - Sequences
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Feb 15th 2010, 06:36 PM
  4. Real Analysis - Sequences
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jan 20th 2009, 02:04 PM
  5. Real Analysis - Sequences #3
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jan 20th 2009, 01:35 PM

Search Tags


/mathhelpforum @mathhelpforum