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Topic: Difficulty with a proof in Baby Rudin
Replies: 4   Last Post: Apr 9, 2013 4:14 PM

 Messages: [ Previous | Next ]
 Paul Posts: 780 Registered: 7/12/10
Re: Difficulty with a proof in Baby Rudin
Posted: Apr 9, 2013 4:14 PM

On Tuesday, April 9, 2013 5:14:25 PM UTC+1, dull...@sprynet.com wrote:
> On Mon, 8 Apr 2013 04:33:13 -0700 (PDT), pepstein5@gmail.com wrote:
>
>
>

> >I have difficulty following the proof of theorem 8.3 in the 3rd edition of Principles of Mathematical Analysis by Rudin. The line I'm struggling with is:
>
> >lim n -> infinity [ sum(i = 1 to infinity) [sum j = 1 to n a_ij ] ] =
>
> >= lim n -> infinity [sum j = 1 to n [sum i = 1 to infinity a_ij ] ]
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> >
>
> >This doesn't seem immediate, and the way I prove this is by theorem 3.55.
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>
>
> From your comments below I gather that 3.55 has something to do with
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> rearranging absolutely convergent series? That has nothing at all to
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> do with this, that I can see. The equality you ask about is entirely
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> triivial.
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>
>
> I don't have the book with me, so I can't say what the section numbers
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> are. But surely one of the very first things he proves about infinite
>
> sums is this:
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>
>
> Lemma. If sum_1^infinity a_j and sum_1^infinity b_j both
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> converge then sum_1^infinity (a_j + b_j) converges and
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> equals sum a_j + sum b_j.
>
>
>
> The lemma is entiirely trivial, has nothing to do with absolute
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> convergence or rearrangements, just the definitions (hint:
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> epsilon = epsilon/2 + epsilon/2).
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>
>
> And it's immediate from the lemma by induction on n that
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>
>
> sum(i = 1 to infinity) [sum j = 1 to n a_ij ] =
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> = sum j = 1 to n [sum i = 1 to infinity a_ij ] .
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>
>
> Hence lim_n of both sides is the same.
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>
>

> >However, the whole statement of 8.3 seems immediate from 3.55 anyway, since you just need to arrange the countable number of terms a_ij into a single sequence and appeal to the absolute convergence.
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> >
>
> >So Rudin's exposition of theorem 8.3 doesn't help me. I can only follow it by using theorem 3.55 and I understand that his intention is to avoid theorem 3.55.
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> >
>
> >What am I missing?
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> >
>
> >Thank You,
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> >
>
> >Paul Epstein

David,

I agree with you, but I resolved the issue myself in my 2nd posting.

Paul

Date Subject Author
4/8/13 Paul
4/8/13 ArtflDodgr
4/9/13 Paul
4/9/13 David C. Ullrich
4/9/13 Paul