## experimentX Group Title Balzano-Weirerstrass Theorem: Does any one know why there is $$2^{m-2}$$ at the $$\mathbb{R^n}$$ while there is $$2^{m-1}$$ on $$\mathbb{R}^1$$? http://i.stack.imgur.com/O14VI.jpg one year ago one year ago

1. experimentX Group Title

$$2^{n-1}$$ the the fifth line from proof while $$2^{m-2}$$ on the right bottom

2. waterineyes Group Title

Wait for @mukushla Ha ha ha ha..

3. ParthKohli Group Title

@experimentX: Apostol?

4. experimentX Group Title

yes!! page no 55

5. ParthKohli Group Title

Which Apostol is it?

6. experimentX Group Title

Tom M. Apostle Mathematical Analysis Second Edition

7. ParthKohli Group Title

I see.

8. joemath314159 Group Title

It looks like its because of the way they labeled the subintervals in their proof. In the one dimensional case, they didnt label the starting interval (the one of length 2a, [-a,a]), while in the n-dimensional case they did (J_1 is the n-dimensional rectangle with sides [-a,a]).

9. experimentX Group Title

the first interval is labelled as [a1,b1] which is [-a, 0] or the other, and it's length is 'a' if we continue this we, we get [an, bn] = a/2^(n-1) for the case of R^1

10. joemath314159 Group Title

right, but in the n dimensional case, the intervals in J_1 have length 2a.

11. ParthKohli Group Title

Am I the only idiot here?

12. experimentX Group Title

J_1 is defined as $$I_1^{(1)} \times I_2^{(1)} \times I_3^{(1)} \times ... I_n^{(1)}$$ where each $$I_k^{(1)}: -a \le x \le 0$$

13. joemath314159 Group Title

14. joemath314159 Group Title

$-a\le x_k\le a$

15. experimentX Group Title

i see i had been looking at the other page.

16. joemath314159 Group Title

yeah, its a little confusing >.<

17. experimentX Group Title

thanks for clearing up!!

18. waterineyes Group Title

@ParthKohli , No you are not alone here..

19. ParthKohli Group Title

lol

20. waterineyes Group Title

Ha ha ha ha.. Even you know about Apostle.. I have heard the word first time in my Life..

21. ParthKohli Group Title

He's just an author yaar

22. waterineyes Group Title

Ha ha ha... I think that is certain set in Mathematics which is given that name.. Ha ha ha. So, I am more Idiot than you..