section_path: Types of limits/In sequences/Metric space/Example: R
section_name: Example: R
paragraph_text: Example: R
  
  An important example is the space of n {\displaystyle n} -dimensional real vectors, with elements x = (x 1, ⋯, x n) {\displaystyle \mathbf {x} =(x_{1},\cdots,x_{n})} where each of the x i {\displaystyle x_{i}} are real, an example of a suitable distance function is the Euclidean distance, defined by d (x, y) = ‖ x − y ‖ = ∑ i (x i − y i) 2. {\displaystyle d(\mathbf {x},\mathbf {y})=\|\mathbf {x} -\mathbf {y} \|={\sqrt {\sum _{i}(x_{i}-y_{i})^{2}}}.} The sequence of points { x n } n ≥ 0 {\displaystyle \{\mathbf {x} _{n}\}_{n\geq 0}} converges to x {\displaystyle \mathbf {x} } if the limit exists and ‖ x n − x ‖ → 0 {\displaystyle \|\mathbf {x} _{n}-\mathbf {x} \|\rightarrow 0}.