Heine Borel Theorem $\iff$ Bolzano Weierstrass Theorem? Are the statements of the Heine-Borel Thm and Bolzano-Weierstrass Thm equivalent?  
 A: See the Wikipedia page for reverse mathematics. Heine-Borel needs the $WKL_0$ axiom set, while Bolzano-Weierstrass needs $ACA_0$. $ACA_0$ is the stronger theory  - all theorems of $WKL_0$ are theorems in $ACA_0$ but not visa versa. There is a base theory for which adding Heine-Borel as an additional axiom yields $WKL_0$, and adding Bolzano-Weierstrass yields $ACA_0$. In particular, B-W is not a theorem in $WKL_0$.
The basic approach of reverse mathematics is to start with a base set of axioms, $RCA_0$, and then find various additional axioms that one could add to this set to get other results. This notion, then, that Heine-Borel is weaker, is a very specific notion, relative to this base theory $RCA_0$.
A: Jupp they are, both are equivalent to have a finite dimensional normed vector space over a complete field. (If you take that every bounded sequence has a convergent subsequence, there are more than one formulation of the bolzano weierstrass theorem).
A: Heine-Borel and Bolzano-Weierstrass 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, stated as:
Theorem A. (Heine-Borel). Every bounded closed subset of R is compact.
Theorem B. (Bolzano-Weierstrass). Every bounded infinite subset of R has a
limit point.
A proof of equivalences of these theorems is given in an article (Classroon Notes), available via the link: http://www.researchgate.net/publication/232863146
(Thanks to Peter for his suggestion.)
