Metric spaces
Metric spaces [ edit ] Definition [ edit ] A point � of the metric space ( � , � ) is the limit of the sequence ( � � ) if: For each real number � > 0 , there is a natural number � such that, for every natural number � ≥ � , we have � ( � � , � ) < � . Symbolically, this is: ∀ � > 0 ( ∃ � ∈ � ( ∀ � ∈ � ( � ≥ � ⟹ � ( � � , � ) < � ) ) ) . This coincides with the definition given for real numbers when � = � and � ( � , � ) = | � − � | . Properties [ edit ] When it exists, the limit of a sequence is unique, as distinct points are separated by some positive distance, so for � less than half this distance, sequence terms cannot be within a distance � of both points. For any continuous function f , if lim � → ∞ � � exists, then lim � → ∞ � ( � � ) = � ( lim � → ∞ � � ) . In fact, a function f is continuous if and only if it preserves the limits of sequences. Cauchy sequences [ edit ] Main article: Cauchy sequence The plot of a Cauchy seque