1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sequence, continuity, connectedness

  1. Sep 28, 2007 #1
    1) Prove that
    lim x_k exsts and find its value if {x_k} is defined by
    x_1 = 1 and x_(k+1) = (1/2) x_k + 1 / (sqrt k)

    [My attempt: Assume the limit exists and equal to L
    then L= (1/2) L + 0
    => (1/2) L = 0
    => L=0
    Now I have to prove that the limit indeed exists, I want to use the monotone sequence theorem (monotone & bounded => convergence), but as I evaluate a few terms, I found that x1<x2, but x2>x3, which makes it not montone...what should I do?]

    Definition: Let S be a subset of R^n. S is called open if it contains none of its boundary points. A point x E R^n is a boundary point of S if every ball centred at x contains both points in S and points in S^c (S complement)

    Definition: A subset of R^n is called compact if it's both closed and bounded.

    Definition: f is continuous at a iff
    lim f(x) = f(a)

    2) Suppose f: R^m->R^n is a map such that for any compact set K C R^n, the preimage set f^(<-)(K) = {x E R^m | f(x) E K} is compact. Is f necessary continuous? Justify.

    3) Suppose f: R^n -> R^k has the following property: For any open set U C R^k, {x| f(x) E U} is an open set in R^n. Show that f is continuous on R^n.

    For #2 and #3, I have absolutely no clue...feeling desperte...

    Any help will be greatly appreciated!:smile:
    Last edited: Sep 28, 2007
  2. jcsd
  3. Sep 28, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    You must show that x is a monotone decreasing sequence. Since x_(k+1) = (1/2) x_k + 1 / (sqrt k), x_(k+1) would be < x_k for sufficiently small values of 1 / (sqrt k). You need to solve for that k. (You can discard the first few terms of the sequence if they are not monotone decreasing.)

    What is your definition of an open set? A compact set?
    Last edited: Sep 28, 2007
  4. Sep 28, 2007 #3


    User Avatar
    Science Advisor
    Homework Helper

    Q1. Perhaps x_2 > x_3 > x_4 > ...?

    Q2. Do you think this is true or false? A continuous function necessarily pulls back closed sets to closed sets. By the Heine-Borel theorem, the compact subsets of R^n (and R^m) are precisely those that are closed and bounded. So f pulls back at least these subsets to closed subsets. But there are more closed sets out there (specifically the ubounded ones). Maybe you can use this to construct a counterexample?

    Q3. What definition of continuity are you working with?
  5. Sep 28, 2007 #4


    User Avatar
    Science Advisor
    Homework Helper

    There is a direct, specific relationship between compact-to-compact and continuity.
  6. Sep 29, 2007 #5
    1) Yes, I've found out that the pattern is x2 > x3 > x4 > x5 > ...

    Now is it possible to show by math induction that x_k > x_(k+1) > 0 (hence bounded and montone) ?

    2) My course didn't cover the Heine-Borel theorem, so I can't use it. How should I do without this theorem?

    3) I have added the definitions used in my textbook...
  7. Sep 29, 2007 #6


    User Avatar
    Science Advisor
    Homework Helper

    #3 is shown more easily by the epsilon-delta definition of continuity. Have you studied that definition?

    #2 is shown more easily using the convergent subsequence definition of compactness; have you studied that definition? Let [itex]\{x_n\}[/itex] be a sequence in the preimage that converges to [itex]x_0[/itex] in the preimage. Do you understand why you can find such a sequence in the preimage?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Sequence, continuity, connectedness