As far as I know, completeness is a property of topological vector space whose topology induced by a metric. While $\mathbb{R}$ is a field, generally speaking, a set, rather than a space. But mathematicians usually say $\mathbb{R}$ is complete (perhaps, with respect to the Euclidean metric $d_E$). Do they exactly mean the metric space $(\mathbb{R},d_E)$ over the field $\mathbb{R}$ or $\mathbb{C}$ is complete?
Furthermore, I read some textbooks regarding the proof of the completeness of real number line that are too complicated, for example here. Can anyone provide me with a simple and clear proof in just a few lines?