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.