Question

Which theorem states that every bounded sequence in \( \mathbb{R}^n \) has a convergent subsequence?

• A. Mean Value Theorem
• B. Bolzano–Weierstrass Theorem
• C. Rolle’s Theorem
• D. Taylor’s Theorem

Hint:

Bounded sequence \( \Rightarrow \) Convergent subsequence (always!)

Answer 1

The Correct Option is B

Concept: Bolzano–Weierstrass Theorem
This is a fundamental theorem in real analysis which states:

• Every bounded sequence in \( \mathbb{R}^n \) has at least one convergent subsequence.

Step 1: Understand bounded sequence
A sequence \( \{x_n\} \) is bounded if: \[ \exists M>0 \text{ such that } \|x_n\| \leq M \text{ for all } n \]
Step 2: Apply the theorem
If the sequence is bounded, then it cannot “escape to infinity,” so there exists a subsequence that converges.
Step 3: Conclusion
\[ \text{Bolzano–Weierstrass Theorem guarantees convergence of a subsequence} \]