Monday, 7 September 2015

real analysis - Proof writing: $sum_{n=1}^{infty}| a_n|


Let n=1an be an absolutely converging series. By definition, this means n=1|an| converges. We want to show that n=1a2n converges absolutely as well.







Here is what I tried:




Let's say n=1|a2n| converges. Let ϵ>0 be given and there exists NN such that



|a2m+1|+|a2m+2|++|a2n|<ϵ for all n>mN. By the triangle inequality,



|a2m+1+a2m+2++a2n||a2m+1|+|a2m+2|++|a2n|



so the Cauchy Criterion guarantees that n=1a2n also converges.







Here is my question:




Is this correct, or do I need to prove anything else?


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