result about arithmetic of convergent sequences

Jan 2016
35
0
United Kingdom
Hello all,

I believe that, given a collection of increasing sequences whose total sum converges, each individual sequence will also converge. Attached is my short proof. Is it correct?

Regards,

Magnitude
 

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v8archie

Math Team
Dec 2013
7,710
2,679
Colombia
Your first line seems to state that all the sequences diverge.

Comments:
  1. You need only prove the result for two sequences. The result for $n$ sequences follows by induction.
  2. The given result then follows by considering that the sum is always greater than each of the series individually. Thus, if one diverges we use the comparison test with that series and the sum.
 
Last edited:
Jan 2016
35
0
United Kingdom
V8archie,

In my first line, I failed to mention the "collection" is a subset of the original bunch. Then I suppose such a collection could diverge, but then their sum diverges and adding all the rest in (which, having not been included in the collection are bounded above) gives that the total diverges, which is a contradiction. Is this sufficient? Meanwhile, I'm trying the induction approach now.
 
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