Understanding the Weierstrass- Bolzano theorem

[chwala]

TL;DR

I am currently looking at this theorem...basically it states that if you have a given sequence that is bounded and infinite then there exists atleast one limit point. Looking at the attached...i would like to know how the author concludes that the subsequence;

${-1,-1,-1,...}$ converges to $-1$.

I guess that should follow from previous step...unless there is a mistake.


Find link here;https://math.libretexts.org/Bookshelves/Analysis/Book%3A_Real_Analysis_(Boman_and_Rogers)/07%3A_Inter

 

[chwala]

Let me pick a sequence, say $a_n=\dfrac{1}{n}$ for example...this is a bounded infinite sequence. I can let my subsequence be defined by;

$b_n= \dfrac{1}{100},\dfrac{1}{200}, \dfrac{1}{300},...$ the limit of this sequence will still tend to $0$.

I am looking at it on the aspect of at least $1$ limit point. Question is, is there another limit point for the given sequence? using the given theorem?

 

[Office_Shredder]

i would like to know how the author concludes that the subsequence;

${-1,-1,-1,...}$ converges to $-1$.

As a standalone claim, this should be exceedingly obvious. If not, apply the definition of a limit without any bells or whistles.