Every real Cauchy sequence \((x_n)_{n\in\mathbb N}\) in the metric space \((\mathbb R,|~|)\) converges against a limit \(a\in\mathbb R\). This is also known as the completeness principle for real numbers.1
fn1. Please note that this proposition is the converse to the corresponding lemma stating that all convergent real sequences are also Cauchy sequences, which is valid for any metric spaces. The converse is valid only for special metric spaces, like \((\mathbb R,|~|)\).
Definitions: 1
Explanations: 2
Proofs: 3 4 5 6 7 8
Propositions: 9
In some bibliography sources (e.g. [^581]), the completeness principle is introduced as an axiom rather than proven (as it is the case in BookOfProofs). At BookOfProofs, one of many axiomatic systems is used for the foundation of real numbers. For more details how this is done at BookOfProofs, please refer to the definition of real numbers. ↩
