The wikipedia proof of Bolzano Weierstrass theorem I was going through the proof that has been written for Bolzano-Weierstrass theorem in the respective Wikipedia page.
http://en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem
I could understand the first part where it assumed the existence of infinite no. of so called peaks but in the part where the finiteness of the peak is assumed I could not understand the logic used after N=n+1. How has it been proven that finite no.of peaks cannot exist or that a convergent sub sequence exists?
Waiting for insightful comments.
 A: This is a pretty cool proof.
In the first part, which you understand, we state that any sequence with infinitely many peaks has a monotonically decreasing subsequence.  In the next part, we show that if there are only finitely many peaks, there has to be a monotonically increasing subsequence.  The key to this next proof is that if a number is not a peak, then there's some greater entry further along in the sequence.  Remember: a member $x_n$ of a sequence is a peak if and only if all of the entries coming afterwards are strictly less than $x_n$.
So, assuming you only have finitely many peaks:
We start with whatever entry comes after the final peak and call this entry $x_{n_1}$.  We know that $x_{n_1}$ is not a peak, since we already hit the last peak.  Because of this, there is some $x_{n_2}$ that is greater than $x_{n_1}$.  $x_{n_2}$ is not a peak, so rinse and repeat and then we have $x_{n_3}$.  We can repeat this process since there are no more peaks to produce an infinite subsequence $\{x_{n_j}\}$ that is monotonically increasing.
I hope that clears things up.
