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!

Prove that the following are equivalent: a) A is bounded, b) A is in a closed ball

  1. Sep 2, 2012 #1
    Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed ball

    1. The problem statement, all variables and given/known data
    The full problem is:
    Let M be a metric space an A[itex]\subseteq[/itex]M be any subset. Prove that the following are equivalent:
    a)A is bounded.
    b)A is contained in some closed ball
    c)A is contained in some open ball.

    I only want help going from A to B, but maybe a little guidance from B to C or A to B--and I will attempt to prove the opposite way.


    2. Relevant equations
    Book definitions:
    A is bounded if [itex]\exists[/itex]R≥0 s.t. d(x,y)≤R [itex]\forall[/itex] x,y[itex]\in[/itex]A
    If a is a nonempty bounded subset of M, the diameter of A is diam(A) = sup{d(x,y):x,y[itex]\in[/itex]A}
    For any x[itex]\in[/itex]M and r>0, the closed ball of radius r around x is [itex]\overline{B}[/itex]r(x)={y[itex]\in[/itex]M:d(y,x)≤R}


    3. The attempt at a solution
    My first thoughts are:
    (=>) A to B
    Spse A is bounded.
    Let R = diam(A)
    [itex]\forall[/itex]x1,x2[itex]\in[/itex]A, d(x1,x2)≤R
    Thus, [itex]\forall[/itex]x[itex]\in[/itex]A, [itex]\exists[/itex]y[itex]\in[/itex]M s.t. d(y,x)≤R
    Let [itex]\overline{B}[/itex]r(x)={y[itex]\in[/itex]M:d(y,x)≤R} be the arbitrary union of y's.
    Thus, [itex]\forall[/itex]x[itex]\in[/itex]A, x[itex]\in[/itex][itex]\overline{B}[/itex]r(x)
    Thus, A[itex]\subseteq[/itex][itex]\overline{B}[/itex]r(x)
     
  2. jcsd
  3. Sep 2, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed

    You're studying Lee's book, aren't you? :smile:

    No, this can't be correct. You have shown that [itex]x\in B_r(x)[/itex]. But here you find for each x, a ball that contains x. So you have a ball for each x in A. You don't want that. You want only one ball that contains all the elements in A. So you want to find an [itex]B_r(x)[/itex] such that [itex]y\in B_r(x)[/itex] for all y in A.

    Now, what if you just take r like you did before, and take x arbitrary (but fixed)??
     
  4. Sep 2, 2012 #3
    Re: Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed

    Lol, I sure am! Brand new edition! I see what you are saying!
    This time I will fix x (since by def of diameter, all x's will be inside that).

    Spse A is bounded,
    Let R = diam(A)
    Let x[itex]\in[/itex]A, [itex]\exists[/itex]y[itex]\in[/itex]M s.t. d(x,y)≤R
    Let [itex]\bar{B}[/itex]r(x) = {y[itex]\in[/itex]M:d(y,x)≤R}
    Thus [itex]\forall[/itex]x[itex]\in[/itex]A, x[itex]\in[/itex][itex]\bar{B}[/itex]r(x) (since [itex]\forall[/itex]x1,x2[itex]\in[/itex]A, d(x1,x2)≤R)
    Thus, A[itex]\subseteq[/itex][itex]\bar{B}[/itex]r(x)
     
  5. Sep 2, 2012 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed

    No, this is just the exact same proof.
    What you want is to fix [itex]x_0[/itex] and R, and prove that
    [itex]\forall y\in A:~y\in \overline{B}_R(x_0)[/itex].
     
  6. Sep 2, 2012 #5
    Re: Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed

    Hmm, I am having a bit of trouble seeing it.
    To be sure, are you saying that the y's need to be in A or can they just be within M and outside of A? Does that make sense?
     
  7. Sep 2, 2012 #6

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed

    You want to prove that all elements in A are in a closed ball. So I pick a closed ball [itex]\overline{B}_R(x_0)[/itex] (with our previous R and [itex]x_0[/itex]).
    Now, I need to prove that A is a subset of this closed ball.

    So I need to prove that for all y in A holds that [itex]d(y,x_0)\leq R[/itex].
     
  8. Sep 3, 2012 #7
    Re: Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed

    I believe that I understand it a bit better. I may have just been fudging up my material a bit.
    For my personal clarity, I want y to be the point within M that satisfy the inequality. We will then label points in x as x1, etc.

    Spse A is bounded
    Let R = diam(A)
    Let x0[itex]\in[/itex]A
    [itex]\exists[/itex]y[itex]\in[/itex]M s.t. d(x0,y)≤R
    Let [itex]\bar{B}[/itex]R(x0) = {y[itex]\in[/itex]M : d(x0,y)≤R}
    Let x1[itex]\in[/itex]A,
    since d(x0,x1)≤R, x1[itex]\in[/itex][itex]\bar{B}[/itex]R(x0)
    Thus, A[itex]\subseteq[/itex][itex]\bar{B}[/itex]R(x0)

    To me this makes sense, by fixing the ball of length R around x0, and because R is = diam (A), we know that if an element is in A then the distance between it and x0 must also be less or equal. Thus, it is within the "area" of the ball.

    How's that?

    Thank you so much for your help!
     
  9. Sep 3, 2012 #8

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Re: Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed

    That's better.

    The point is that all points in M satisfy the inequality.

     
  10. Sep 3, 2012 #9
    Re: Prove that the following are equivalent: a) A is bounded, b) A is "in" a closed

    Thank you very much! I see what I was doing wrong now. I love this site! lol
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook