Let \((X,d)\) be a metric space and \((x_n)\) a sequence on \(X\). The sequence converges to some \(x\in X\) if and only if for all \(\varepsilon>0\) there is a \(n_0\in \mathbb{N}\) such that \(d(x_n,x)<\varepsilon\) for all \(n\ge n_0\).
Let \((X,d)\) be a metric space and \((x_n)\) a sequence on \(X\). The sequence converges to some \(x\in X\) if and only if for all \(\varepsilon>0\) there is a \(n_0\in \mathbb{N}\) such that \(d(x_n,x)<\varepsilon\) for all \(n\ge n_0\).