Proof that if unordered sum is finite, then there are finitely many positive terms

Asked Mar 03 '21 at 13:48

Active Mar 03 '21 at 14:02

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I'm confused about a statement I've seen.

For $\{c_{i} : i \in I\}$, and $P = \{i \in I : c_{i} > 0\}$ it states:

$\sum_{i \in I} c_{i} < \infty \implies $P countable

I understand this is true intuitively but I don't know how you'd formulate the proof for it.

edited Mar 03 '21 at 14:02

asked Mar 03 '21 at 13:48

mathPhys

2

This is related – Peter Melech Mar 03 '21 at 13:52
Is $u_i=c_i$ ?? – Fred Mar 03 '21 at 13:53
@mathPhys a nice proof is given in the question to the above link – Peter Melech Mar 03 '21 at 13:55
@lulu $I$ is some indexing set. $P$ is the set of strictly positive elements of ${c_{i} : i \in I}$ – CyclotomicField Mar 03 '21 at 13:56
I've corrected my mistake now @Fred. – mathPhys Mar 03 '21 at 14:03
Thanks @PeterMelech thats very helpful – mathPhys Mar 03 '21 at 14:03
@mathPhys You're welcome! – Peter Melech Mar 04 '21 at 12:28

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