A Understanding the Weierstrass- Bolzano theorem

  • A
  • Thread starter Thread starter chwala
  • Start date Start date
  • Tags Tags
    Theorem
chwala
Gold Member
Messages
2,827
Reaction score
415
TL;DR Summary
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
1666813869721.png
 
Physics news on Phys.org
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?
 
chwala said:
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.
 
  • Like
Likes topsquark and chwala
Back
Top