Mathematics.

sequences and convergence

Cauchy Sequences

Real Analysis35 minDifficulty4 out of 10

You should know: sequences and limits

Overview

A sequence (aₙ) of real numbers is Cauchy if its terms eventually become arbitrarily close to each other, not merely close to some fixed limit. Formally, for every ε > 0 there is an index N beyond which any two terms aₘ, aₙ (both indexed at least N) satisfy |aₘ − aₙ| < ε. The remarkable fact, due essentially to Cauchy and made rigorous by Weierstrass and Cantor, is that in the real numbers a sequence converges if and only if it is Cauchy — so one can verify convergence without ever knowing the limit in advance. This equivalence is exactly the completeness property of ℝ, and it fails over the rationals, where a Cauchy sequence of rationals (e.g. successive decimal truncations of √2) can fail to converge to a rational limit.

Intuition

Instead of asking 'do the terms approach some fixed target L?' — which requires already knowing or guessing L — the Cauchy condition asks 'do the terms eventually huddle together, getting closer to each other as the sequence progresses?' Picture the terms as a swarm of points on the real line: being Cauchy means that past some point N, the whole tail of the swarm fits inside an arbitrarily small interval, even though you never named where that interval is centered. In ℝ this internal huddling is enough to guarantee a limit exists (because ℝ has no 'holes'); in ℚ the swarm can huddle around an irrational hole with no rational number there to converge to.

Formal Definition

Definition

A sequence (aₙ) of real numbers is called a Cauchy sequence if:

∀ε>0, ∃N∈N such that ∀m,n≥N, ∣am−an∣<ε\forall \varepsilon > 0,\ \exists N \in \mathbb{N} \text{ such that } \forall m, n \ge N,\ |a_m - a_n| < \varepsilon

Terms eventually cluster arbitrarily close to each other, with no reference to a limit value

Cauchy criterion
(an) converges in R  ⟺  (an) is Cauchy(a_n) \text{ converges in } \mathbb{R} \iff (a_n) \text{ is Cauchy}
Cauchy completeness of ℝ
Every Cauchy sequence is bounded.\text{Every Cauchy sequence is bounded.}
Boundedness (necessary condition)

Worked Examples

  1. For m, n ≥ N, bound the difference using the triangle inequality and 1/n ≤ 1/N.

    ∣am−an∣=∣1m−1n∣≤1m+1n≤2N|a_m - a_n| = \left|\frac{1}{m} - \frac{1}{n}\right| \le \frac{1}{m} + \frac{1}{n} \le \frac{2}{N}

  2. Force 2/N < ε by choosing N > 2/ε (exists by the Archimedean property).

    N>2εN > \frac{2}{\varepsilon}

  3. Then for all m, n ≥ N, the bound gives the Cauchy condition.

    ∣am−an∣≤2N<ε|a_m - a_n| \le \frac{2}{N} < \varepsilon

Answer: Choosing N > 2/ε shows (1/n) is Cauchy; e.g. for ε = 0.01, N = 201 suffices.

Practice Problems

Difficulty 4/10

Find N (in terms of ε) proving aₙ = (3n+1)/n is Cauchy, using |aₘ − aₙ| directly.

Difficulty 3/10

Which property is a necessary consequence of a sequence being Cauchy?

Difficulty 6/10

Prove that every convergent sequence of real numbers is Cauchy.

Quiz

The Cauchy criterion for a sequence (aₙ) states:
Which of the following is true about Cauchy sequences of rational numbers?

Summary

  • A sequence is Cauchy if ∀ε>0 ∃N such that ∀m,n≥N, |aₘ−aₙ|<ε — terms cluster near each other, not necessarily near a known limit.
  • In ℝ, a sequence converges if and only if it is Cauchy; this equivalence is the Cauchy completeness of the real numbers.
  • ℚ is not complete: Cauchy sequences of rationals can fail to converge to a rational number (e.g. decimal truncations of √2).

References

  1. WebsiteWikipedia — Cauchy sequence