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I'm currently working through introductory functional analysis from kreyszig, and I don't quite understand one of the proofs. this is it:

proof

My question is why it is necessary to go from $k = 1, 2, ...$ to the infinite sum. Why can't we just let $n\to\infty$ directly in (3)?

user2520938
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    It is not clear what alternative you are proposing to the cited argument, "Why can't we just let $n\to \infty$ directly"? Even if there were an alternative approach, the proof given in the cited text seems clear. Do you really have a specific question about a step taken in this proof? – hardmath Nov 24 '14 at 19:32
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    Well I think it's pretty clear what I'm asking here. I'm wondering why we can't go directly from (3) to (5) without going through the sums for $k = 1, 2, ...$. I agree that the proof is fairly clear, but without understanding why one of the steps is actually necessary I feel like I don't fully understand it. – user2520938 Nov 24 '14 at 19:38
  • Perhaps the clarity you seek would be better addressed by your giving the argument that you would use, and then we can judge whether it succeeds in jumping straight from (3) to (5) "directly", if I'm following you. – hardmath Nov 24 '14 at 19:48
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    If $(a_n^{(k)}){k\in\mathbb{N}}$ is a convergent sequence of non-negative real numbers for every $n\in \mathbb{N}$, then it does not follow without further assumptions that $$\lim{k\to\infty} \sum_{n=0}^\infty a_n^{(k)} = \sum_{n=0}^\infty \lim_{k\to\infty} a_n^{(k)}.$$ – Daniel Fischer Nov 24 '14 at 19:49
  • @DanielFischer Thanks for the answer. After thinking a bit about this, it does indeed answer my question. – user2520938 Nov 24 '14 at 20:01

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