Bolzano-Weierstrass Theorem

Start with a bounded sequence (aₙ) inside [A, B]. Halve to [A, (A+B)/2] and [(A+B)/2, B]; one half contains infinitely many terms — call it I₁. Pick aₙ₁ ∈ I₁. Halve I₁; one sub-half contains infinitely many of the remaining terms — call it I₂. Pick aₙ₂ ∈ I₂ with n₂ > n₁. Iterate. The intervals Iₖ have lengths halving each step, so they nest down to a single point L. The chosen subsequence (aₙₖ) lies in Iₖ, hence |aₙₖ − L| ≤ length(Iₖ) → 0, so aₙₖ → L. The bisection method is the standard constructive proof and a useful template for many compactness arguments.

In ℓ² the standard basis {eₙ} is bounded (||eₙ|| = 1) but pairwise distance ||eₘ − eₙ|| = √2 means no subsequence is Cauchy. By completeness, no subsequence converges. The phenomenon is that infinite dimensions allow sequences to spread out arbitrarily without piling up. Riesz's lemma generalizes: the closed unit ball is compact iff the space is finite-dimensional. To get compactness back in infinite dimensions you switch to a weaker topology (weak compactness, Banach-Alaoglu) or impose extra conditions like equicontinuity (Arzelà-Ascoli).

The completeness axioms of ℝ are interderivable. Cauchy completeness (every Cauchy sequence converges), nested-intervals (∩[aₙ, bₙ] is non-empty), least-upper-bound (every bounded set has a sup), and Bolzano-Weierstrass (every bounded sequence has a convergent subsequence) are all equivalent characterizations. The bisection proof of B-W relies on the nested-intervals property; Cauchy completeness in turn follows from B-W (a Cauchy sequence is bounded, has a convergent subsequence, and its Cauchy property forces the whole sequence to share that limit). They are five faces of the same coin.

Extreme value theorem: continuous f : [a, b] → ℝ attains its max. Pick a sequence (xₙ) ⊂ [a, b] with f(xₙ) → sup f([a, b]); the supremum exists because f([a, b]) is a bounded set in ℝ (continuous image of bounded with a bound from compactness arguments). The xₙ's are bounded (in [a, b]), so by Bolzano-Weierstrass there is a subsequence xₙₖ → x* in [a, b] (closedness). By continuity f(xₙₖ) → f(x*), and the limit is the supremum. So f(x*) = max f. The same argument proves the infimum is attained.

A set K is sequentially compact if every sequence in K has a subsequence that converges to a point in K. In ℝⁿ this is equivalent to compactness (Heine-Borel) and to closed-and-bounded. In general metric spaces sequential compactness is equivalent to compactness (Bolzano-Weierstrass-like proofs use total boundedness). In general topological spaces, compactness and sequential compactness diverge; one neither implies the other in full generality. For practical analysis on ℝⁿ or any metric space, sequential compactness is the working notion of compactness.

Arzelà-Ascoli (1883/1895) characterizes precompact subsets of C(K, ℝ) — continuous functions on a compact set with the sup norm — as bounded and equicontinuous families. It is the function-space analog of Bolzano-Weierstrass: in finite dimensions, bounded ⇒ has convergent subsequence; in C(K, ℝ), bounded + equicontinuous ⇒ has uniformly convergent subsequence. This is the key step in proving ODE solutions exist (Peano), in Picard iteration convergence, and in many PDE compactness arguments where you extract limit functions from approximation sequences.