Cauchy Sequences and Completeness: Explained with Examples

A sequence is Cauchy when its terms get arbitrarily close to each other — not to any known target. Formally, beyond some point all terms lie within ε of one another. In ℝ, every Cauchy sequence converges, and this equivalence is exactly what completeness means.

This matters because it lets you prove convergence without knowing the limit in advance — the typical situation with infinite sums. It also exposes the hole in ℚ: Cauchy sequences of rationals can aim at √2, which is missing there.

Key Concepts

• Cauchy criterion: terms eventually cluster together, with no limit mentioned.
• Complete space: a space where every Cauchy sequence converges inside the space.
• ℝ is complete: every Cauchy sequence of reals has a real limit.
• ℚ is incomplete: Cauchy sequences of rationals can converge "to" an irrational gap.
• Convergent ⇒ Cauchy: always true; the converse needs completeness.

Formulas

• Cauchy definition: ∀ε > 0 ∃N such that m, n > N ⇒ |a_m − a_n| < ε.
• Completeness of ℝ: every Cauchy sequence in ℝ converges to a real number.
• Series test: Σaₖ converges iff its partial sums form a Cauchy sequence.
• Contraction link: |aₙ₊₁ − aₙ| shrinking geometrically forces the Cauchy property.

Worked Example

Approximate √2 by Newton's iteration: x₁ = 1, xₙ₊₁ = (xₙ + 2/xₙ)/2. Compute: x₂ = (1 + 2)/2 = 1.5; x₃ = (1.5 + 2/1.5)/2 = (1.5 + 1.3333)/2 ≈ 1.41667; x₄ = (1.41667 + 2/1.41667)/2 ≈ (1.41667 + 1.41176)/2 ≈ 1.4142157. The gaps shrink fast: |x₃ − x₂| ≈ 0.0833, |x₄ − x₃| ≈ 0.00245 — the terms are clustering together, the Cauchy property in action. The limit satisfies L = (L + 2/L)/2, giving L² = 2, so L = √2 ≈ 1.41421356; check x₄² ≈ 2.00000006. Every xₙ is rational, yet the sequence has no rational limit — a Cauchy sequence in ℚ with nothing to converge to. In ℝ the limit exists: that is completeness.

Common Misconceptions

• Cauchy and convergent are the same by definition. Convergent always implies Cauchy, but Cauchy implies convergent only in a complete space.
• Cauchy sequences must tend to 0. No — the constant sequence 5, 5, 5, … is Cauchy and converges to 5.
• Terms decreasing to 0 is enough. The harmonic partial sums have differences → 0 yet diverge; the Cauchy condition is about all pairs of late terms, not neighbours.

Real-World Applications

• Proving infinite series converge before computing their sums numerically.
• Constructing ℝ itself: reals can be defined as equivalence classes of Cauchy sequences of rationals.
• Fixed-point algorithms in machine learning rely on Cauchy-type contraction arguments.

Cauchy sequences capture "settling down" without naming a destination, and completeness guarantees the destination exists. Together they are the engine behind every convergence proof for series and iterations.