Scope of the theorem and its limits

Moving to higher dimensions

In $\mathbb{R}^d$, the theorem remains true, and is proved by successive extractions. Let $\left(u_n\right)$ be a bounded sequence in $\mathbb{R}^2$, with coordinates $\left(x_n, y_n\right)$.

The sequence $(x_n)$ is bounded: we extract a convergent $\left(x_{\varphi(n)}\right)$ from it. The sequence $\left(y_{\varphi(n)}\right)$ is still bounded: we extract a convergent $\left(y_{\varphi \circ \psi(n)}\right)$ from it. Since a subsequence of a convergent sequence converges to the same limit, the first coordinate still converges after this second extraction. The subsequence $\left(u_{\varphi \circ \psi(n)}\right)$ therefore converges in $\mathbb{R}^2$.

The procedure is repeated $d$ times in dimension $d$, and applies as is to $\mathbb{C}$, viewed as $\mathbb{R}^2$.

The major theorems that follow from it

Result	Role of Bolzano-Weierstrass
Extreme value theorem	extract from a maximizing sequence
Heine's theorem	extract from the two sequences of the negation
$\mathbb{R}$ is complete	a bounded Cauchy sequence with a convergent subsequence converges
Heine-Borel theorem in $\mathbb{R}^d$	equivalence of the definitions of compactness

Take the first one. If $f$ is continuous on $[a, b]$, consider $M = \sup f$ — a priori in $\mathbb{R} \cup \{+\infty\}$ — and a sequence $(x_n)$ in $[a, b]$ such that $f(x_n) \to M$. The sequence $(x_n)$ is bounded, so we extract $x_{\varphi(n)} \to c$, with $c \in [a, b]$ since the interval is closed. Continuity gives $f\left(x_{\varphi(n)}\right) \to f(c)$, hence $M = f(c)$: the supremum is finite and attained.

Where the theorem stops

In infinite dimension, it fails. In the space $\ell^2$ of square-summable sequences, consider the canonical basis $\left(e_n\right)$, where $e_n$ has a $1$ in position $n$ and $0$s elsewhere. This sequence is bounded, since $\lVert e_n \rVert = 1$ for all $n$, and yet

$$\lVert e_n - e_m \rVert = \sqrt{2} \qquad \text{whenever } n \neq m$$

No subsequence is Cauchy, so none converges. This is the content of Riesz's theorem: the closed unit ball of a normed vector space is compact if and only if the space is finite-dimensional.

What the theorem says about $\mathbb{R}$

Bolzano-Weierstrass is not a computational accident: it is equivalent to the least upper bound property, and hence to the very construction of $\mathbb{R}$. In $\mathbb{Q}$, the statement is false — the sequence of decimal truncations of $\sqrt{2}$ is bounded, and none of its subsequences converges to a rational number.

$$\text{least upper bound} \iff \text{nested intervals} \iff \text{Bolzano-Weierstrass} \iff \text{Cauchy sequences converge}$$

Classic pitfall

Concluding "the sequence converges" after applying the theorem. It only gives the existence of a convergent subsequence. To get back to the whole sequence, an additional argument is needed: uniqueness of the cluster point, monotonicity, or the Cauchy property.

Second pitfall: applying the theorem without checking that the sequence is bounded. For $u_n = n$, no subsequence converges; for $u_n = n \cos n$, which is unbounded, the theorem allows no conclusion.