Results 1 to 2 of 2

Math Help - convergence of a sequence wrt a metric

  1. #1
    Junior Member
    Joined
    Nov 2005
    Posts
    31

    convergence of a sequence wrt a metric

    (1) Let [0,1]* be the set of all functions N-->[0,1] (ie of all sequences in [0,1]). Define a sequence (Fn) in [0,1]* by:

    Fn(k)= 0 if k different from n; 1 otherwise.

    Consider the following metric on [0,1]*: d1(f,g):=sup{|f(n)-g(n)|:n element of N}.
    Determine whether (Fn) converges with respect to d1.

    (2) same as in (1) but with respect to the metric d2(f,g):= (sum from i=0 to infinity)[(2^-i)|f(i)-g(i)|.

    Note that N is the set of all natural numbers including zero.

    I do not know how to check convergence with respect to a metric.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jun 2005
    Posts
    295
    Awards
    1
    Convergence of a sequence (x_n) implies that it satisfies the Cauchy condition, namely \forall \epsilon>0\ \exists N = N(\epsilon) \cdot m,n>N \Rightarrow d(x_m,x_n) < \epsilon. (Note that the Cauchy criterion does not imply convergence unless the metric space is complete.)

    Now consider (1). We have d(f_m,f_n) = 1 whenever m \neq n. Hence the Cauchy criterion with  \epsilon = 1/2 does not hold.

    On the other hand, convergence of a sequence x_n to y is defined as saying that the sequence of real numbers d(x_n,y) converges to zero.

    For (2), you have to decide what the sequence converges to, and I suggest you consider the sequence z_n = 0. Then consider d(f_n,z) = 2^{-n} to see that f_n \rightarrow z.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cauchy Sequence in Metric Space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 8th 2011, 09:29 AM
  2. Cauchy sequence convergence in a general metric space
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: November 29th 2010, 09:31 PM
  3. Cauchy Sequence in a Metric Space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 8th 2010, 03:05 AM
  4. metric space: convergent sequence
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 11th 2009, 11:40 AM
  5. Replies: 1
    Last Post: January 16th 2009, 01:00 PM

Search Tags


/mathhelpforum @mathhelpforum