# Bolzano-Weierstrass Theorem

## Theorem

Every bounded sequence of real numbers has a convergent subsequence.

### General Form

Every infinite bounded space in a real Euclidean space has at least one limit point.

## Proof 1

Let $\sequence {x_n}$ be a bounded sequence in $\R$.

By the Peak Point Lemma, $\sequence {x_n}$ has a monotone subsequence $\sequence {x_{n_r} }$.

Since $\sequence {x_n}$ is bounded, so is $\sequence {x_{n_r} }$.

Hence, by the Monotone Convergence Theorem (Real Analysis), the result follows.

$\blacksquare$

## Proof 2

Let $\sequence {x_n}_{n \mathop \in \N}$ be a bounded sequence in $\R$.

By definition there are real numbers $c, C \in \R$ such that $c < x_n < C$.

Then at least one of the sets:

- $\set {x_n: c < x_n < \dfrac {c + C} 2 }, \set {x_n : \dfrac {c + C} 2 < x_n < C }, \set {x_n : x_n = \dfrac {c + C} 2 }$

contains infinitely many elements.

If the set $\set {x_n : x_n = \dfrac {c + C} 2 }$ is infinite there is nothing to prove.

If this is not the case, choose the first element from the set, say $x_{k_1}$.

Repeat this process for $\sequence {x_n}_{n \mathop > k_1}$.

As a result we obtain subsequence $\sequence {x_{k_n} }_{n \mathop \in \N}$.

By construction $\sequence {x_{k_n} }_{n \mathop \in \N}$ is a Cauchy sequence and therefore converges.

$\blacksquare$

## Also see

## Source of Name

This entry was named for Bernhard Bolzano and Karl Weierstrass.

## Historical Note

The Bolzano-Weierstrass Theorem is a crucial property of the real numbers discovered independently by both Bernhard Bolzano and Karl Weierstrass during their work on putting real analysis on a rigorous logical footing.

It was originally referred to as **Weierstrass's Theorem** until Bolzano's thesis on the subject was rediscovered.

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Exercise $5$