Let $d: \mathbb{R} \times \mathbb{R} \to [0,\infty)$ be the metric defined by $d(x,y)=|x-y|$. Then the [metric space](/page/Metric%20Space) $(\mathbb{R},d)$ is complete; equivalently, every [Cauchy sequence](/page/Cauchy%20Sequence) $(x_n)_{n\in\mathbb{N}}$ in $(\mathbb{R},d)$ converges to some $x\in\mathbb{R}$.