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experimentX

  • 2 years ago

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

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  1. experimentX
    • 2 years ago
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    \( 2^{n-1} \) the the fifth line from proof while \( 2^{m-2} \) on the right bottom

  2. waterineyes
    • 2 years ago
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    Wait for @mukushla Ha ha ha ha..

  3. ParthKohli
    • 2 years ago
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    @experimentX: Apostol?

  4. experimentX
    • 2 years ago
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    yes!! page no 55

  5. ParthKohli
    • 2 years ago
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    Which Apostol is it?

  6. experimentX
    • 2 years ago
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    Tom M. Apostle Mathematical Analysis Second Edition

  7. ParthKohli
    • 2 years ago
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    I see.

  8. joemath314159
    • 2 years ago
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    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
    • 2 years ago
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    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
    • 2 years ago
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    right, but in the n dimensional case, the intervals in J_1 have length 2a.

  11. ParthKohli
    • 2 years ago
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    Am I the only idiot here?

  12. experimentX
    • 2 years ago
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    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
    • 2 years ago
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  14. joemath314159
    • 2 years ago
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    \[-a\le x_k\le a\]

  15. experimentX
    • 2 years ago
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    i see i had been looking at the other page.

  16. joemath314159
    • 2 years ago
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    yeah, its a little confusing >.<

  17. experimentX
    • 2 years ago
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    thanks for clearing up!!

  18. waterineyes
    • 2 years ago
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    @ParthKohli , No you are not alone here..

  19. ParthKohli
    • 2 years ago
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    lol

  20. waterineyes
    • 2 years ago
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    Ha ha ha ha.. Even you know about Apostle.. I have heard the word first time in my Life..

  21. ParthKohli
    • 2 years ago
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    He's just an author yaar

  22. waterineyes
    • 2 years ago
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    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..

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