Bolzano–Weierstrass Theorem
Every bounded sequence has a convergent subsequence.
Key idea
Every bounded sequence has a convergent subsequence.
In depth
Every bounded sequence of real numbers has a convergent subsequence. Proof idea: repeatedly bisect the interval and keep a half containing infinitely many terms. It is a compactness property of closed intervals.
Worked example
The bounded sequence sin n has a convergent subsequence, although sin n itself does not converge.
Real-world uses
- Guaranteeing that a bounded sequence of measurements has a convergent part
- Compactness proofs
Common mistake
Applying it to unbounded sequences.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Bolzano–Weierstrass Theorem?
Every bounded sequence has a convergent subsequence. It is a topic at the expert level in Analysis & Topology.
Where is Bolzano–Weierstrass Theorem used in real life?
Real-world examples: Guaranteeing that a bounded sequence of measurements has a convergent part; Compactness proofs.
What should I learn before Bolzano–Weierstrass Theorem?
Make sure you are comfortable with Subsequences, the previous topic in Analysis & Topology.
What is a common mistake with Bolzano–Weierstrass Theorem?
Applying it to unbounded sequences.
How can I practise Bolzano–Weierstrass Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Bolzano–Weierstrass Theorem?
Move on to Cauchy Sequences next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.