---
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: 2026-09-12
---

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# 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 its late terms bunch together: past some N, every pair sits within any given tolerance. Showing terms get pairwise close, without naming a limit, proves a sequence is Cauchy. That is the point of the criterion: it certifies convergence while the limit is still unknown. For 1 over n, tolerance 0.01 needs N equal to 100.

**Example.** The approximations 1, 1.4, 1.41, 1.414, and so on bunch beautifully while chasing root 2 from inside the rationals. The sequence is Cauchy in the rationals yet has no rational limit. In the reals, completeness closes that gap, so Cauchy and convergent mean the same thing.

Bolzano-Weierstrass says every bounded real sequence has a convergent subsequence. The alternating (-1) to the n diverges, yet its even indexed terms sit constant at 1. Extracting such a subsequence is a skill: halve the trap, keep the infinite half, and repeat. Partial sums show the same idea when disturbances fade, as with 1 plus 1 over 2 plus 1 over 4 plus 1 over 8 reaching 1.875.

**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)