1. ## Bolzano–Weierstrass theorem

I am looking for the most compact proof for the Bolzano–Weierstrass theorem,anybody have any links/ideas. Thanks.

2. Originally Posted by MathSucker
I am looking for the most compact proof for the Bolzano–Weierstrass theorem,anybody have any links/ideas. Thanks.
Some statements of this theorem vary. But here is one source.

3. What do you think of this one? I want to be able to write it out in the shortest time possible. Is there anything I can trim out?

If $(x_n)$ is a sequence in the closed segment $[a,b]$ then there exists a subsequence of $(x_n)$ that converges to a limit in $[a,b]$.

Let $c$ be an arbitrary point in $[a,b]$, then $[a,c]$ can contain and infinite number of members of $(x_n)$ and it can contain a finite number of members of $(x_n)$.

Let: $D=\{c\in[a,b] \, : \,[a,c]$ contains a finite number of members of $(x_n)\}$

If $c$ is in $D$, then $[a,c]$ has a finite number of members of $(x_n)$, and any subsest of $[a,c]$ can only contain a finite number of members of $(x_n)$. Hence, for any $c \in D$, all the points between $a$ and $c$ are also in $D$.

Let $x=sup(D)$ and assume $x \in (a,b)$.

Take an arbitrarily small $\varepsilon$. $[a,x+ \varepsilon]$ is not in $D$ since $x+ \varepsilon$ is greater than $sup(D)$. $[a, x - \varepsilon]$ is in $D$ since $x - \varepsilon$ is less than $sup(D)$.

$[a, x - \varepsilon]$ is a subset of $[a, x + \varepsilon]$. The larger set contains an infinite number of members of $(x_n)$ and the subset contains a finite number of members of $(x_n)$. So, the complement of the subset contains an infinite number of members of $(x_n)$.

Thus, every $x$ in $[a,b]$, has a neighbourhood that contains an infinite number of members of $(x_n)$.

Set $\varepsilon=1$. Take any member of $(x_n)$ in $(x-1,x+1)$ and make this the first member of a sequence.

Set $\varepsilon=1/2$. Take any member of $(x_n)$ in $(x-1/2,x+1/2)$ with a higher index than the previous member, and make this the next member of the sequence.

Set $\varepsilon=1/4$ and so forth indefinitely.

This results in a subsequence of $(x_n)$ that converges to $x$.

4. The best proof based on what? It is, in fact, possible to take "Bolzano-Weierstrasse" as a defining axiom for the real numbers and then the other basic properties (monotone convergence, upper bound property, Heine-Borel, etc.) can be proved from that. But you can also prove "Bolzano-Weierstrasse" from any of them.

5. There are very short proofs, but they may use machinery/terms above the level of your current course. What level is this at?

6. Undergraduate level, but we're given zero worked solutions. The prof basically just writes out proofs on the board, hands us a list of questions, and then walks out. I was just looking for a proof for the "Bolzano-Weierstrasse" theorem that I could bang out on the exam in as little time possible. I found that one, and then wrote it out in my own words and was wondering if it made sense.