## You are here

Homeproof of Bolzano-Weierstrass Theorem

## Primary tabs

# proof of Bolzano-Weierstrass Theorem

To prove the Bolzano-Weierstrass theorem, we will first need two lemmas.

Lemma 1.

proof.

Let $(s_{n})$ be a bounded, nondecreasing sequence. Let $S$ denote the set $\{s_{n}:n\in\mathbb{N}\}$. Then let $b=\sup S$ (the supremum of $S$.)

Choose some $\epsilon>0$. Then there is a corresponding $N$ such that $s_{N}>b-\epsilon$. Since $(s_{n})$ is nondecreasing, for all $n>N$, $s_{n}>b-\epsilon$. But $(s_{n})$ is bounded, so we have $b-\epsilon<s_{n}\leq b$. But this implies $|s_{n}-b|<\epsilon$, so $\lim s_{n}=b$. $\square$

(The proof for nonincreasing sequences is analogous.)

Lemma 2.

Every sequence has a monotonic subsequence.

proof.

First a definition: call the $n$th term of a sequence *dominant* if it is greater than every term following it.

For the proof, note that a sequence $(s_{n})$ may have finitely many or infinitely many dominant terms.

First we suppose that $(s_{n})$ has infinitely many dominant terms. Form a subsequence $(s_{{n_{k}}})$ solely of dominant terms of $(s_{n})$. Then $s_{{n_{{k+1}}}}<s_{{n_{k}}}$ $k$ by definition of “dominant”, hence $(s_{{n_{k}}})$ is a decreasing (monotone) subsequence of ($s_{n}$).

For the second case, assume that our sequence $(s_{n})$ has only finitely many dominant terms. Select $n_{1}$ such that $n_{1}$ is beyond the last dominant term. But since $n_{1}$ is not dominant, there must be some $m>n_{1}$ such that $s_{m}>s_{{n_{1}}}$. Select this $m$ and call it $n_{2}$. However, $n_{2}$ is still not dominant, so there must be an $n_{3}>n_{2}$ with $s_{{n_{3}}}>s_{{n_{2}}}$, and so on, inductively. The resulting sequence

$s_{1},s_{2},s_{3},\ldots$ |

is monotonic (nondecreasing). $\square$

proof of Bolzano-Weierstrass.

The proof of the Bolzano-Weierstrass theorem is now simple: let $(s_{n})$ be a bounded sequence. By Lemma 2 it has a monotonic subsequence. By Lemma 1, the subsequence converges. $\square$

## Mathematics Subject Classification

40A05*no label found*26A06

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia