How can we prove Bolzano – Weierstrass theorem which says, "In $\Bbb R$ , every bounded, infinite subset has at least one limit point (in $\Bbb R$) " by using Heine- Borel theorem (In $\Bbb R$, every open cover of a closed and bounded subset has a finite subcover)
[Math] How to prove Bolzano-Weierstrass theorem by using Heine -Borel theorem
real-analysis
Best Answer
If $A$ is bounded and infinie subset of $R$ , it is obviously in a closed and bounded set like $B$ so by the $Heine-Borel \ \ theorem$ every open cover of $B$ has a finite subcover. suppose that $A$ does not have any limit point at $B$ so every point of $A$ like $t$ is contained in a open ball that it's intersection with $B$ is only ${t}$ . now these open balls create a open covering for $B$ that has finite subcover. now $A$ is contained in the union of finite balls that every ball has only one element of $A$ so $A$ must be finite set and it is contradiction.