TITLE: Difference between Heine-Borel Theorem and Bolzano-Weierstrass Theorem
QUESTION [17 upvotes]: It's a very basic (may be a trivial) question but what is the exact difference, if any, between Heine Borel Theorem and Bolzano Weierstrass  Theorem. It is true that one (Heine Borel) can be proved from another (Bolzano Weierstrass ).
Heine Borel Theorem: Subspace of $\mathbb{R}^n$ is compact iff it is closed and bounded.
Bolzano Weierstrass  Theorem: Every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence.

REPLY [2 votes]: The Heine-Borel and the Bolzano-Weierstrass theorems are two fundamental
results in real analysis. These theorems are equivalent in the sense that their proofs can be derived from each other. In fact, there are other axioms and results such as completeness axiom, the nested interval property, the Dedekind cut axiom of continuity and Cauchy’ s general principle of convergence which are equivalent to these theorems. Most textbooks do not mention these equivalences.
Apparently, it is not within the scope of elementary textbooks on real analysis to include proofs of all these equivalences. Among these results, the Heine-Borel theorem and the Bolzano-Weierstrass theorem are of fundamental importance in applications and generalization to a wider framework of topological spaces.
For detail, see
http://www.researchgate.net/publication/232863146