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Math Help - Limits of Converging Sequences

  1. #1
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    Limits of Converging Sequences

    I just want a clear proof for this theorem.

    Theorem. The closure of a set S in R^n is the set of all limits of converging sequences of points from S.

    Thanks!!
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  2. #2
    Senior Member Tinyboss's Avatar
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    The closure of S in R^n is S unioned with the set of limit points of S, i.e. points x such that any neighborhood of x contains some point of S other than x itself. The exercise is to show that this condition on x is equivalent to being the limit of a convergent sequence of points in S.

    Hint: every neighborhood of a point contains an epsilon-ball.
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