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Math Help - Sequences and closed sets

  1. #1
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    Sequences and closed sets

    Let A \doteq \{x_n\}^\infty_{n \doteq 1} be a sequence and let B be the set of all subsequential limits of A. I have to show that A \cup B is closed.

    I got that B is closed from that theorem in Rudin's textbook that said that the subsequential limits form a closed set which leads that B is closed. Now I figured that I should prove that A is closed as well in order to complete the proof, and that's where I'm stuck at.
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    Quote Originally Posted by darkchibi07 View Post
    Let A \doteq \{x_n\}^\infty_{n \doteq 1} be a sequence and let B be the set of all subsequential limits of A. I have to show that A \cup B is closed.
    I got that B is closed from that theorem in Rudin's textbook that said that the subsequential limits form a closed set which leads that B is closed. Now I figured that I should prove that A is closed as well in order to complete the proof, and that's where I'm stuck at.
    You cannot prove that A is closed, consider A = \left\{ {\frac{1}{n}} \right\}.
    But if a is a limit point of A them some subsequence of A converges to a. Because B is the set of all subsequential limits of A then it follows that A \cup B is closed. And of course B is closed because any point limit point of B is also a limit point of A, so it must belong to B meaning it is closed.
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  3. #3
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    Was the subsequence of A you chose just arbituary, or do I have to construct it formally?
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