5.1. Open and Closed Sets

Theorem 5.1.8: Closed Sets, Accumulation Points, and Sequences

  • A set S R is closed if and only if every Cauchy sequence of elements in S has a limit that is contained in S.

  • Every bounded, infinite subset of R has an accumulation point.

  • If S is closed and bounded, and is any sequence in S, then there exists a subsequence of that converges to an element of S.


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