Exercise 1193Topology › Topology of ℝⁿL1Added 3 Nov 2026

Sequential compactness in $\mathbb{R}^n$

#Bolzano-Weierstrass#sequential compactness

Statement

State the Bolzano–Weierstrass theorem in $\mathbb{R}^n$ .

Hint

Every bounded sequence has a convergent subsequence.

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Theorem. Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence.

Link to compactness. If $K$ is compact, every sequence in $K$ has a subsequence converging in $K$ (sequential compactness).

Conclusion. Compactness $\Leftrightarrow$ sequential compactness in $\mathbb{R}^n$ .