Linked from

The 16 pages that link to Bolzano–Weierstrass theorem, each with the reason it gives.

• SequenceRelated: It guarantees convergent structure inside every bounded sequence of real vectors.

• Cauchy criterionCompared with: It guarantees a convergent subsequence, not convergence of the whole sequence.

• Intermediate value theoremRelated: Compactness arguments alongside the intermediate value theorem establish extrema and related existence results.

• Extreme value theoremRelated: A maximizing sequence and a convergent subsequence give a standard proof on compact subsets of Euclidean space.

• Arzelà–Ascoli theoremRelated: Its finite-dimensional subsequence principle underlies the theorem’s diagonal argument.

• Heine–Borel theoremRelated: It explains why boundedness yields convergent subsequences, while closedness keeps their limits inside the set.

• Bernard BolzanoBroader topic: This theorem bears Bolzano’s name and exemplifies his foundational work on mathematical analysis.

• Sequential compactnessBroader topic: It gives the standard Euclidean example of sequential compactness.

• Banach–Alaoglu theoremCompared with: The theorem asserts topological compactness, not necessarily a convergent-subsequence property for arbitrary nets or sequences.

• Upper boundRelated: Bounds on sequences enable a compactness result through convergent subsequences.

• Completeness of the real numbersRelated: Compactness arguments using this theorem depend on completeness of the real numbers.

• Weierstrass theoremRelated: A maximizing sequence yields a convergent subsequence whose limit preserves the function value.

• Axiom of countable choiceRelated: Its usual nested-interval proof makes successive selections, illustrating countable choice in analysis.

• Monotone convergence theoremCompared with: It guarantees subsequential convergence without requiring the entire sequence to be monotone.

• Heine–Cantor theoremRelated: It is a classical compactness criterion underlying sequential proofs of the theorem.

• Cantor's intersection theoremRelated: Its subsequence compactness principle underlies familiar metric-space proofs.