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Math Help - Open and Closed Sets

  1. #1
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    Open and Closed Sets

    (1)
    Let B= \{\frac{(-1)^{n}n}{n+1}: n =1,2,3,...\}

    (a) Find the limit points of B.
    (b) Is B a closed set?
    (c) Is B an open set?
    (d) Does B contain any isolated points?
    (e) Find \overline{B}

    (For (e), use this definition. Given a set A \subseteq R, let L be the set of all limit points of A. The closure of A is defined to be \overline{A}= A \cup L

    (2) Decide wheather the following sets are open, closed, or neither. If a set is not open, find a point in the set for which there is no \epsilon-neighborhood contained in the set. If a set is not closed, find a limit point that is not contained in the set.

    (a) Q-rational numbers
    (b) N-natural numbers
    (c) \{x\in R: x>0\}
    (d) (0,1]= \{x\in R: 0<x\underline{<}1\}
    (e) {1+1/4+1/9+...+1/ n^2: n \in N}

    (3) Let a \in A. Prove that a is an isolated point of A if and only if there exists an \epsilon-neighborhood V_{\epsilon}(a) such that V_{\epsilon}(a) \cap A= {a}.


    Any and all help would be greatly appreciated, thanks!
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  2. #2
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    Sorry, is this the right forum for this type of question? This is from my analysis class. Thanks.
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  3. #3
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    I have solved (1) and (2), so if anyone could take a crack at #3 that would be coolio!
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