section_path: Types of limits/Limit sets/Limit set of a sequence
section_name: Limit set of a sequence
paragraph_text: Limit set of a sequence
  
  Let { a n } n > 0 {\displaystyle \{a_{n}\}_{n>0}} be a sequence in a topological space X {\displaystyle X}. For concreteness, X {\displaystyle X} can be thought of as R {\displaystyle \mathbb {R} }, but the definitions hold more generally. The limit set is the set of points such that if there is a convergent subsequence { a n k } k > 0 {\displaystyle \{a_{n_{k}}\}_{k>0}} with a n k → a {\displaystyle a_{n_{k}}\rightarrow a}, then a {\displaystyle a} belongs to the limit set. In this context, such an a {\displaystyle a} is sometimes called a limit point.
  
  A use of this notion is to characterize the "long-term behavior" of oscillatory sequences. For example, consider the sequence a n = (− 1) n {\displaystyle a_{n}=(-1)^{n}}. Starting from n=1, the first few terms of this sequence are − 1, + 1, − 1, + 1, ⋯ {\displaystyle -1,+1,-1,+1,\cdots }. It can be checked that it is oscillatory, so has no limit, but has limit points { − 1, + 1 } {\displaystyle \{-1,+1\}}.