## 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**.