In coursebook's proof: Let $(x_n)$ be limited sequence.
Then first is created subsequence $$a_k = sup\lbrace x_n:n\geq k\rbrace$$
So we are choosing first supremum of whole sequence, and then supremum of sequence which is whole sequence minus sequence up to point of latest supremum, et cetera.
At this point sequence $a_k$ is limited and decreasing, thus it is convergent and when $k\rightarrow \infty$ we have $\lim(a_k) = \inf (a_k) \equiv a$
Then from here it's worked out to from, where theorem is proved by principle I couldn't find english translation for, and which I think is irrelevant.
But for the actual question, why can't I, instead of supremums of the parts, choose the actual peaks? So that sequence $a_k$ would consist only elements of original sequence, and giving convergent subsequence right away.