Home > Analysis > Dini’s Theorem

Dini’s Theorem


Let a<b be two real numbers and f_n :[a,b ]\to \Bbb{R} a sequence of  continuous functions which converge pointwise to a continuous function f.

  1. Dini’s Theorem. Suppose that the sequence (f_n) has the property that f_n \leq f_{n+1}. Prove that the convergence f_n \to f is uniform.
  2. Suppose that every function f_n is increasing. Prove that the convergence f_n \to f is uniform.
  3. If every function f_n is convex on [a,b], prove that the convergence f_n \to f is uniform on every compact interval in (a,b).
Advertisements
Categories: Analysis Tags:
  1. chandrasekhar
    February 24, 2011 at 8:07 am

    I don’t understand the third problem. convergence f_{n} \to f is uniform on every compact in (a,b). what does that mean

    • February 24, 2011 at 10:41 am

      It is a typo. T should have written compact interval…

      It means that the convergence may not be uniform on [a,b], but for any [c,d] \subset [a,b] the convergence is uniform.

  2. Markus
    March 11, 2011 at 2:41 pm

    Dinis theorem needs the continuity of f !! See f_n(x)=x^n over [0,1], for example.

    • March 11, 2011 at 8:24 pm

      You are right. I corrected the statement of the problem.

  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: