Bolzano–Weierstrass Theorem

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A bounded sequence keeps returning to a cluster point

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Sequencesin(n)sin(n) + 0.45 cos(2.3n)(−1)ⁿ (1 + 1/n)1 / nCluster point L

0.50

Radius ε

0.10

Terms N

80

Bounded sequence aₙ

Press Play to reveal the terms, or move L and ε.

|aₙ| ≤ M and |aₙ − L| < ε for infinitely many n

Boundedness prevents a sequence from escaping to infinity, but it does not force the whole sequence to settle. This theorem supplies the next-best guarantee: some infinite subsequence does converge.

Theorem for sequences

Let $(x_n)$ be a bounded sequence in the finite-dimensional Euclidean space $\mathbb{R}^d$, where $d\geq 1$. Then $(x_n)$ has a convergent subsequence. The limit lies in $\mathbb{R}^d$; it need not be one of the terms of the sequence.

Boundedness is the only hypothesis on the sequence. In particular, its set of terms need not be closed.

Proof

Write $x_n=(x_n^{(1)},\ldots,x_n^{(d)})$. Since $(x_n)$ is bounded in the Euclidean norm, each coordinate sequence $(x_n^{(j)})$ is bounded in $\mathbb{R}$.

We first recall the one-dimensional argument. A bounded real sequence lies in some closed interval $I_0$. Bisect that interval. At least one of the two closed halves contains terms with infinitely many indices; choose such a half as $I_1$. Continue, each time bisecting the chosen interval and retaining a closed half that contains terms with infinitely many indices. This gives nested closed intervals

$$I_0\supseteq I_1\supseteq I_2\supseteq\cdots$$

whose lengths tend to zero. By the nested interval theorem their intersection consists of one real number $a$. Choose indices $n_k$ strictly increasing so that the $k$th selected term lies in $I_k$. For every $m\geq k$, both the $m$th selected term and $a$ lie in $I_k$, so their distance is at most the length of $I_k$. Hence the selected subsequence converges to $a$.

Apply this real result first to the first coordinate of $(x_n)$, obtaining a subsequence along which that coordinate converges. Apply it to the second coordinate of this subsequence, then to the third, and so on. There are only finitely many coordinates, so after the final step there is one subsequence $(x_{n_k})$ along which every coordinate converges, say $x_{n_k}^{(j)}\to a^{(j)}$ for $1\leq j\leq d$. (Passing to a subsequence at a later step preserves convergence already obtained.)

Let $a=(a^{(1)},\ldots,a^{(d)})$. For any $\varepsilon>0$, coordinate convergence gives, for all sufficiently large $k$ and every $j$, $|x_{n_k}^{(j)}-a^{(j)}|<\varepsilon/\sqrt d$. Therefore

$$\|x_{n_k}-a\|_2 =\sqrt{\sum_{j=1}^d|x_{n_k}^{(j)}-a^{(j)}|^2}<\varepsilon.$$

Thus $x_{n_k}\to a$ in $\mathbb{R}^d$.

Compactness of subsets

The sequence theorem should not be confused with a statement that every bounded subset is compact. For a subset $K\subseteq\mathbb{R}^d$, the Heine–Borel theorem says that $K$ is compact if and only if it is both closed and bounded. Equivalently, in Euclidean space, $K$ is compact if and only if every sequence in $K$ has a subsequence converging to a point of $K$ (sequential compactness).

Closedness matters in this set formulation: for example, $(0,1)$ is bounded but not compact, and the sequence $x_n=1/n$ in it has no subsequence converging to a point of $(0,1)$. The Bolzano–Weierstrass theorem itself only asserts that a bounded sequence in $\mathbb{R}^d$ has a subsequence converging in $\mathbb{R}^d$.

A quick example and why it matters

The bounded sequence $x_n=(-1)^n$ does not converge, but its even-indexed subsequence is constantly $1$ and its odd-indexed subsequence is constantly $-1$. This illustrates both the guarantee and its limited scope: the theorem promises at least one convergent subsequence, not a unique limit for the original sequence. It is a basic tool for extracting convergent subsequences when proving existence of limits or compactness results.

Takeaway

In finite-dimensional Euclidean space, boundedness guarantees a convergent subsequence in the ambient space. Compactness of a set adds the essential requirement that limits remain in the set.

Reference

• Bolzano–Weierstrass theorem