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Math Help - Prove that the closed interval [0,1] is a closed set

  1. #1
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    Prove that the closed interval [0,1] is a closed set

    Prove that the closed interval [0,1] is a closed set and that the open interval (0,1) is an open set..

    Could I just say that for [0,1], every open ball B(0,r), r > 0 contains at least one point less than 0 and therefore not an element of [0,1] to prove it is closed

    and for (0,1), its complement (-inf, 0] U [1, inf) must be closed by similar reasoning?



    is there a way to incorporate the theorem:

    a set S is closed iff for all sequences {xk} such that xk is an element of S for each k and xk --> x, it is the case that x is an element of S


    I feel like Im basically just stating the definition of closed/open sets up top, and I don't know if that's an acceptable way to do it

    Thanks
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    Re: Prove that the closed interval [0,1] is a closed set

    Quote Originally Posted by JaredG View Post
    Prove that the closed interval [0,1] is a closed set and that the open interval (0,1) is an open set..
    Could I just say that for [0,1], every open ball B(0,r), r > 0 contains at least one point less than 0 and therefore not an element of [0,1] to prove it is closed
    and for (0,1), its complement (-inf, 0] U [1, inf) must be closed by similar reasoning?
    There are as many ways to prove this as there are people reading this.
    If a<0 then a\in\left( { - \infty ,\frac{a}{2}} \right) and \left( { - \infty ,\frac{a}{2}} \right)\cap [0,1]=\emptyset
    A simpler statement if b>1 proving the complement of [0,1] is open.
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    Re: Prove that the closed interval [0,1] is a closed set

    Thank you for responding but I cant make sense of that..

    I was asking if the method I was suggesting would be sufficient.
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    Re: Prove that the closed interval [0,1] is a closed set

    Quote Originally Posted by JaredG View Post
    Thank you for responding but I cant make sense of that..
    I was asking if the method I was suggesting would be sufficient.
    I am sorry about that.
    I guess that make us even because I can't make sense of what you posted.
    But then I have only taught this material for forty years.

    Do you know how to prove sets of real numbers are open or closed?
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    Re: Prove that the closed interval [0,1] is a closed set

    Sorry I'll try to write it more formally.

    Using this definition of an open set: A set S in Rn is open if for every element x in S, there is an r > 0 such that B(x,r) is a subset of S

    suppose [0,1] is open

    [0,1] contains the element 0

    B(0,r) contains (0 - r) which is not an element of [0,1] for any r > 0

    therefore B(x,r) is not a subset of S for all x in S, which implies [0,1] must be closed by contradiction


    Do you know how to prove sets of real numbers are open or closed?
    No, I thought that's what was being asked here though?



    Im interested in the proofs you suggested, could you explain them to someone who hasn't taught the material for 40 yrs? I don't understand how that intersection being empty says anything about [0,1] being closed. I could replace [0,1] with (0,1) and the statement would still be true..
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