section_path: Types of limits/In sequences/Metric space
section_name: Metric space
paragraph_text: Metric space
  
  The discussion of sequences above is for sequences of real numbers. The notion of limits can be defined for sequences valued in more abstract spaces, such as metric spaces. If M {\displaystyle M} is a metric space with distance function d {\displaystyle d}, and { a n } n ≥ 0 {\displaystyle \{a_{n}\}_{n\geq 0}} is a sequence in M {\displaystyle M}, then the limit (when it exists) of the sequence is an element a ∈ M {\displaystyle a\in M} such that, given ϵ > 0 {\displaystyle \epsilon >0}, there exists an N {\displaystyle N} such that for each n > N {\displaystyle n>N}, we have d (a, a n) < ϵ. {\displaystyle d(a,a_{n})<\epsilon.} An equivalent statement is that a n → a {\displaystyle a_{n}\rightarrow a} if the sequence of real numbers d (a, a n) → 0 {\displaystyle d(a,a_{n})\rightarrow 0}.