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.

Context