--- title: "Cauchy Sequences and the Bolzano-Weierstrass Theorem" description: "Terms that eventually crowd together converge, in the reals but not in the rationals. Every bounded sequence has a convergent subsequence, which is the workhorse behind existence proofs." canonical: https://lightmysky.com/learn/mathematics/cauchy-sequences-and-the-bolzano-weierstrass-theorem-mt_nreDylVkSU source: https://lightmysky.com/learn/mathematics/cauchy-sequences-and-the-bolzano-weierstrass-theorem-mt_nreDylVkSU.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Cauchy Sequences and the Bolzano-Weierstrass Theorem Terms that eventually crowd together converge, in the reals but not in the rationals. Every bounded sequence has a convergent subsequence, which is the workhorse behind existence proofs. Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/cauchy-sequences-and-the-bolzano-weierstrass-theorem-mt_nreDylVkSU ## Ready when they can - Show a sequence is Cauchy without naming its limit - Give a Cauchy sequence of rationals with no rational limit - Extract a convergent subsequence from a bounded sequence ## Lesson: Bunching up without a target A sequence is Cauchy when late terms bunch together. Formally, for every positive epsilon there is an integer N such that for all m and n greater than or equal to N, the distance between a_m and a_n is less than epsilon. No limit appears in this definition. For 1 over n and tolerance 0.01, choose N equal to 101: two terms past that cutoff differ by at most 1 over 101, which is below 0.01. Cauchy sequences are bounded too. Use epsilon 1: every term after one cutoff lies within 1 of a fixed tail term, and the finitely many earlier terms can be covered by one larger bound. **Example.** The approximations 1, 1.4, 1.41, 1.414, and so on are rational decimal truncations that chase root 2. Once k decimal places are fixed, the truncation is within 10 to the negative k of root 2, so two sufficiently late truncations differ by less than any chosen tolerance. They form a Cauchy sequence in the rationals, but their real limit is irrational, so no rational limit exists. In the reals, completeness closes that gap: Cauchy and convergent mean the same thing. Bolzano-Weierstrass says every bounded real sequence has a convergent subsequence. To extract one, trap the sequence in a closed interval, bisect the interval, keep a half containing infinitely many terms, and repeat. The nested intervals point to a limit for a chosen subsequence. The alternating (-1) to the n diverges, yet its even indexed terms stay constant at 1. Geometric partial sums show the same bunching idea: 1 plus 1 over 2 plus 1 over 4 plus 1 over 8 equals 1.875, and later additions get smaller. **Recap.** Cauchy means late terms bunch, completeness turns bunching into convergence, and boundedness always hides a convergent subsequence. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [The Completeness Axiom: Suprema and Infima](https://lightmysky.com/learn/mathematics/the-completeness-axiom-suprema-and-infima-mt_xjI-pIfh95) - [Convergence of Sequences, Rigorously](https://lightmysky.com/learn/mathematics/convergence-of-sequences-rigorously-mt_yB5M-DCH8V) ## Opens up - [Completeness and the Contraction Mapping Theorem](https://lightmysky.com/learn/mathematics/completeness-and-the-contraction-mapping-theorem-mt_DYsLzg-IaK) - [Continuity and Uniform Continuity](https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz) - [Normed Spaces, Completeness and Banach Spaces](https://lightmysky.com/learn/mathematics/normed-spaces-completeness-and-banach-spaces-mt_y_jueKnovX) - [Compactness and Connectedness](https://lightmysky.com/learn/mathematics/compactness-and-connectedness-mt_ybV1lUedHs)