Mathematics

The Bolzano-Weierstrass Theorem: Every Bounded Sequence Has a Convergent Subsequence

We prove the Bolzano-Weierstrass theorem — that every bounded sequence in Euclidean space has a convergent subsequence — and explore its central role in real analysis as a bridge between boundedness and convergence.

Analysis

The Theorem

Bolzano-Weierstrass Theorem

Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence.

That is: if $(x_k)_{k=1}^{\infty} \subset \mathbb{R}^n$ and there exists $M > 0$ with $\|x_k\| \leq M$ for all $k$ , then there exist indices $k_1 < k_2 < k_3 < \cdots$ such that $(x_{k_j})$ converges.

Intuition

If infinitely many points are crammed into a bounded region, they must "pile up" somewhere. The accumulation point is the limit of a convergent subsequence.

This is the sequential version of compactness: a set is compact if and only if every sequence in it has a convergent subsequence. The Bolzano-Weierstrass theorem is equivalent to the statement that closed bounded subsets of $\mathbb{R}^n$ are compact.

Proof for $\mathbb{R}$ (Bisection Method)

Proof.

Let $(x_k)$ be a bounded sequence in $\mathbb{R}$ with $x_k \in [a, b]$ for all $k$ .

Step 1 — Bisect. Divide $[a, b]$ into two halves: $[a, m]$ and $[m, b]$ where $m = (a+b)/2$ . At least one half contains infinitely many terms of the sequence. Call it $[a_1, b_1]$ and choose an index $k_1$ with $x_{k_1} \in [a_1, b_1]$ .

Step 2 — Iterate. Bisect $[a_1, b_1]$ and choose a half $[a_2, b_2]$ containing infinitely many terms, picking $k_2 > k_1$ with $x_{k_2} \in [a_2, b_2]$ .

Continue to get nested intervals:

$[a, b] \supset [a_1, b_1] \supset [a_2, b_2] \supset \cdots$

with $b_j - a_j = (b - a)/2^j$ and $x_{k_j} \in [a_j, b_j]$ .

Step 3 — Convergence. Both sequences $(a_j)$ and $(b_j)$ are monotone and bounded, hence convergent. Since $b_j - a_j \to 0$ , they converge to the same limit $L$ :

$a_j \leq x_{k_j} \leq b_j \implies x_{k_j} \to L$

by the squeeze theorem. $\square$

Proof for $\mathbb{R}^n$

Proof (component-wise extraction)

Let $(x_k) = ((x_k^{(1)}, \ldots, x_k^{(n)}))$ be bounded in $\mathbb{R}^n$ .

Step 1. The first components $(x_k^{(1)})$ are bounded in $\mathbb{R}$ . By the $\mathbb{R}$ case, extract a subsequence $(x_{k_j^{(1)}})$ along which $x^{(1)}$ converges.

Step 2. Along this subsequence, the second components are still bounded. Extract a further subsequence along which $x^{(2)}$ also converges.

Step $i$ . Continue for each component. After $n$ extractions, we have a subsequence along which all $n$ components converge, hence the subsequence converges in $\mathbb{R}^n$ . $\square$

Alternative Proof via Monotone Subsequences

There is a beautiful alternative using the following lemma:

Lemma (Monotone Subsequence). Every sequence of real numbers has a monotone subsequence (either non-decreasing or non-increasing).

Proof of the Lemma. Call index $k$ a peak if $x_k \geq x_j$ for all $j > k$ .

If there are infinitely many peaks $k_1 < k_2 < \cdots$ , then $x_{k_1} \geq x_{k_2} \geq \cdots$ is non-increasing.
If there are only finitely many peaks, let $N$ be larger than all peak indices. Starting from $N$ , $x_N$ is not a peak, so there exists $k_1 > N$ with $x_{k_1} > x_N$ . Then $k_1$ is not a peak, so there exists $k_2 > k_1$ with $x_{k_2} > x_{k_1}$ . Continuing, we get $x_{k_1} < x_{k_2} < \cdots$ , which is strictly increasing. $\square$

Now, a bounded monotone sequence converges (the Monotone Convergence Theorem), giving Bolzano-Weierstrass.

Equivalent Formulations

In $\mathbb{R}^n$ , the following are equivalent:

Bolzano-Weierstrass: Every bounded sequence has a convergent subsequence.
Heine-Borel: Closed bounded sets are compact.
Completeness: Every Cauchy sequence converges.
Least Upper Bound Property: Every nonempty bounded-above set has a supremum.
Nested Intervals: Nested closed intervals with lengths tending to zero have a common point.

These are all equivalent characterizations of the completeness of $\mathbb{R}$ . Each can be derived from any other.

Applications

Extreme Value Theorem

Theorem. A continuous function $f: K \to \mathbb{R}$ on a compact set $K$ attains its supremum.

Proof using BW. Let $M = \sup\{f(x) : x \in K\}$ . Choose $(x_k) \subset K$ with $f(x_k) \to M$ . By Bolzano-Weierstrass, $(x_k)$ has a convergent subsequence $x_{k_j} \to x^* \in K$ (since $K$ is closed). By continuity, $f(x^*) = M$ . $\square$

Existence of Limits

The theorem is used constantly to extract convergent subsequences from bounded approximating sequences. This is the standard technique for proving existence in:

Optimization problems (direct method of calculus of variations)
Fixed-point theorems
Solutions to differential equations

The Arzelà-Ascoli Theorem

Bolzano-Weierstrass for sequences of functions: if a sequence of functions is uniformly bounded and equicontinuous, it has a uniformly convergent subsequence. The proof uses a diagonal argument applied to BW.

Failure in Infinite Dimensions

In infinite-dimensional spaces, Bolzano-Weierstrass fails. In $\ell^2$ , the standard basis vectors $e_1, e_2, e_3, \ldots$ satisfy $\|e_k\| = 1$ (bounded) but $\|e_j - e_k\| = \sqrt{2}$ for $j \neq k$ , so no subsequence is Cauchy, let alone convergent.

This failure motivates the study of weak compactness: the unit ball in a reflexive Banach space is weakly sequentially compact (every bounded sequence has a weakly convergent subsequence).

Historical Notes

Bernard Bolzano (1817) proved that a continuous function changing sign on an interval must have a zero — the intermediate value theorem — using what amounted to the BW property. Karl Weierstrass systematically developed the theory of limits and subsequences in his Berlin lectures in the 1860s.

The theorem was a key step in the arithmetization of analysis — the program of placing calculus on rigorous foundations, replacing geometric intuition with precise definitions of limits, continuity, and convergence.

Limsup and Liminf

Every bounded sequence $(x_k)$ in $\mathbb{R}$ has a largest and smallest subsequential limit:

$\limsup_{k \to \infty} x_k = \inf_{n \geq 1} \sup_{k \geq n} x_k, \qquad \liminf_{k \to \infty} x_k = \sup_{n \geq 1} \inf_{k \geq n} x_k$

Bolzano-Weierstrass guarantees these are achieved: there exists a subsequence converging to $\limsup$ and another converging to $\liminf$ . The sequence converges if and only if $\limsup = \liminf$ .

Summary

\begin{aligned} &(x_k) \text{ bounded in } \mathbb{R}^n \\[6pt] &\Downarrow \text{ Bolzano-Weierstrass} \\[6pt] &\exists\, \text{subsequence } x_{k_j} \to L \\[6pt] &\Updownarrow \\[6pt] &\text{Closed bounded sets in } \mathbb{R}^n \text{ are compact (Heine-Borel)} \\[6pt] &\text{Fails in infinite dimensions} \implies \text{weak compactness needed} \end{aligned}

References

Rudin, W., Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.
Abbott, S., Understanding Analysis, 2nd edition, Springer, 2015.
Bartle, R. and Sherbert, D., Introduction to Real Analysis, 4th edition, Wiley, 2011.
Wikipedia — Bolzano-Weierstrass theorem
Wikipedia — Compact space
MIT OpenCourseWare — Real Analysis

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