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

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Registered: 7/12/10
Difficulty with a proof in Baby Rudin
Posted: Apr 8, 2013 7:33 AM
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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 ] ]

This doesn't seem immediate, and the way I prove this is by theorem 3.55. 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.

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.

What am I missing?

Thank You,

Paul Epstein

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