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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 \(L\): \(\forall \epsilon\gt0:\exists N\gt0:\forall n\gt N:d(L,a_n)\lt\epsilon\). 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.