Exercise 1887Topology › Topology of ℝL1Added 3 Nov 2026

Sequential compactness in $\mathbb{R}$

#sequential compactness#Bolzano-Weierstrass

Statement

State the Bolzano–Weierstrass theorem and explain its link with compactness of $[0,1]$ .

Hint

Every bounded sequence has a convergent subsequence.

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Solution

Theorem (Bolzano–Weierstrass). Every bounded sequence of reals has a convergent subsequence.

Link. If $K\subseteq\mathbb{R}$ is compact (closed and bounded), every sequence in $K$ is bounded, so has a convergent subsequence; its limit lies in $K$ since $K$ is closed.

Conclusion. Compactness implies sequential compactness in $\mathbb{R}$ ; the converse also holds.

Sequential compactness in $\mathbb{R}$

Statement

Solution

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