<-- Go Back Last Updated: 11/01/2025

Limit (in a Metric Space)**

Given a metric space \((X,d)\) and a sequence \((a_i)_{i=1}^n\) in \(X\), a point \(L\in X\) is a limit of the sequence if, for any given radius, there is a point in the sequence beyond which all terms lie within that distance of \(L\): \(\forall \epsilon>0:\exists N>0:\forall n>N:d(L,a_n)<\epsilon\). If a limit exists then it is unique, and it is also the limit in the topological sense when \(X\) is endowed with the metric topology.