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
- 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.