Cauchy Sequence

It lets you prove a sequence converges without first knowing the limit. The standard ε-N convergence definition requires |aₙ − L| < ε — but L is exactly what you don't know in advance. Cauchy replaces the unknown L with a comparison between two sequence terms |aₘ − aₙ| < ε, both of which exist as soon as the sequence does. In ℝ (and any complete space), Cauchy implies convergent, so you get convergence for free once you verify mutual closeness. This is the standard tool for showing series converge, integrals exist, and fixed-point iterations terminate.

A metric space (X, d) is complete if every Cauchy sequence in X converges to a point of X. ℝ is complete — a foundational fact often taken as the defining axiom of the real numbers. ℂ is complete. Any closed subset of a complete space is complete. Banach spaces are complete normed vector spaces; Hilbert spaces are complete inner-product spaces. Incompleteness is the absence of certain limits: ℚ is not complete because some Cauchy sequences (like rational approximations of π or √2) have limits that aren't rational.

Take the set of all Cauchy sequences of rational numbers. Declare two sequences (aₙ), (bₙ) equivalent when (aₙ − bₙ) → 0. The equivalence classes form ℝ — each real number is identified with the set of all rational Cauchy sequences that should converge to it. Addition and multiplication descend term-by-term to operations on classes. The completeness of ℝ follows by construction: any Cauchy sequence of these classes itself converges to another class. This is one of two standard constructions; the other is Dedekind cuts. The Cauchy construction generalizes: completing any metric space by adding equivalence classes of Cauchy sequences yields a complete metric space containing the original densely.

Bolzano-Weierstrass says every bounded sequence in ℝⁿ has a convergent subsequence. Combined with the fact that Cauchy sequences are bounded, this lets you extract a convergent subsequence from any Cauchy sequence. Once one subsequence converges, the Cauchy property forces the whole sequence to converge to the same limit. This is the standard short proof that ℝⁿ is complete. In abstract metric spaces, Bolzano-Weierstrass (sequential compactness of bounded closed sets) and completeness become independent properties.

Convergent always implies Cauchy: if aₙ → L, then |aₘ − aₙ| ≤ |aₘ − L| + |L − aₙ| can be made arbitrarily small by triangle inequality. The reverse holds only in complete spaces. In ℚ, the rational truncations of π form a Cauchy sequence (terms are eventually within 10⁻ᵏ) but no rational number serves as their limit — convergence fails. In a general metric space, the existence of "holes" where Cauchy sequences pile up without a limit is what completeness rules out.

Given any metric space (X, d), the completion (X̂, d̂) is the unique (up to isometry) complete metric space in which X embeds as a dense subset. Construction mirrors ℝ-from-ℚ: take Cauchy sequences in X, identify those that should have the same limit, define distance between equivalence classes as the limit of pairwise distances. Examples: ℂ((t)) (formal Laurent series) is the completion of ℂ[t, t⁻¹] under a t-adic metric; the p-adic numbers ℚₚ are the completion of ℚ under the p-adic absolute value; L²([0,1]) is the completion of continuous functions under the L² norm.