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Math Help - Derived Set Proof

  1. #1
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    Exclamation Derived Set Proof

    Hello, I am currently studying for my analysis midterm, and I can't seem to figure out how to solve the following:

    Let S,TRn. Show that (SUT)' = S'UT'.

    Where S' and T' represent derived sets.

    Any help with this problem would be greatly appreciated!
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  2. #2
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    Quote Originally Posted by Majialin View Post
    Hello, I am currently studying for my analysis midterm, and I can't seem to figure out how to solve the following:

    Let S,TRn. Show that (SUT)' = S'UT'.

    Where S' and T' represent derived sets.

    Any help with this problem would be greatly appreciated!
    x is an accumulation point of (SUT)' iff there exists a sequence of distinct points a_n of SUT with |x - a_n| < 1/n. This is true iff either infinitely many of the points belong to S or infinitely many belong to T (or both), which is true iff either x belongs to S' or x belongs to T'. Hence (SUT)' = S'UT'.
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  3. #3
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    You have posted this question twice. That is strictly against forum rules.
    But I will respond this way. There is only one way that the implication is problematic.
    That is to prove that \left( {S \cup T} \right)^\prime   \subseteq S' \cup T'.
    If x is limit point of S\cup T and x is limit point of S then we are done.

    But if and x is not limit point of S you must prove that x must a limit point of t.
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