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Math Help - How does one prove something is closed and bounded?

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    How does one prove something is closed and bounded?

    I'm wondering how one goes about proving something is both closed and bounded. I need to prove that a closed ball of radius r about x0 is closed and bounded. I'm just not sure how these proofs typically go, and I don't have a great definition of closed or bounded. Thanks for any help.
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    Re: How does one prove something is closed and bounded?

    What are your definitions of closed and bounded?
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    Re: How does one prove something is closed and bounded?

    There are a ton of ways to show a set is closed. Here are some:
    1) Show the set contains all its limit points.
    2) Show that the complement of the set is open.
    3) Show it's a finite union, or arbitrary intersection, of other closed sets.
    4) Show it equals its closure.
    5) Show it's the pre-image of a closed set.
    6) Show it's the image under a closed map of a closed set.
    7) Perhaps it's known to be a compact subset of a Hausdorf space?
    And I'm sure there are plenty more. Which to try depends on the circumstances.

    If it's already been proven that the distance function in a metric space is a continuous function, then there's a very quick proof:
    (Note that the distance function on R^n is obviously continuous, and so this works in R^n)
    Let f = dist( *, x0):V -> R. Then D = f-inverse([0, r0]).
    So since f is continous, and [0, r0] is known to be closed in R, have that D is closed.

    I think that, maybe in this case, the next most simple way is to prove that D-complement is open.
    Pick x not in D (that closed bal). Seek to show that x is in an open set that doesn't intersect D.
    Since x not in D, dist(x, x0) > r. Let a = dist(x, x0) - r.
    Let U = { x' | dist(x', x) < a/2}.
    Then U open (why), x in U, and y in U implies y not in D (why? hint: triangle inequality).
    So D complement is open. Therefore D is closed.
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    Re: How does one prove something is closed and bounded?

    Quote Originally Posted by Shanter View Post
    I'm wondering how one goes about proving something is both closed and bounded. I need to prove that a closed ball of radius r about x0 is closed and bounded. I'm just not sure how these proofs typically go, and I don't have a great definition of closed or bounded. Thanks for any help.
    Let \mathcal{B}=\{x:d(x,x_0)\le r\}. If t\not\in\mathcal{B} then
    \alpha=d(t,x_0)-r>0.
    Now \mathcal{O}=\{x:d(x,t)< \alpha\} is an open set such that t\in\mathcal{O}~\&~\mathcal{O}\cap\mathcal{B}= \emptyset.
    That shows \mathcal{B}^c is open. So \mathcal{B} is closed.

    If u\in\mathcal{B}~\&~v\in\mathcal{B} show that d(u,v)\le 2r.
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