# proof of Bolzano-Weierstrass Theorem

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

Lemma 1.

All bounded monotone sequences converge.

proof.

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

Choose some $\u03f5>0$. Then there is a corresponding $N$ such that ${s}_{N}>b-\u03f5$. Since $({s}_{n})$ is nondecreasing, for all $n>N$, ${s}_{n}>b-\u03f5$. But $({s}_{n})$ is bounded, so we have $$. But this implies $$, so $lim{s}_{n}=b$. $\mathrm{\square}$

(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 $$ $k$ 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

$${s}_{1},{s}_{2},{s}_{3},\mathrm{\dots}$$ |

is monotonic (nondecreasing). $\mathrm{\square}$

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. $\mathrm{\square}$

Title | proof of Bolzano-Weierstrass Theorem |
---|---|

Canonical name | ProofOfBolzanoWeierstrassTheorem |

Date of creation | 2013-03-22 12:22:26 |

Last modified on | 2013-03-22 12:22:26 |

Owner | akrowne (2) |

Last modified by | akrowne (2) |

Numerical id | 5 |

Author | akrowne (2) |

Entry type | Proof |

Classification | msc 40A05 |

Classification | msc 26A06 |