Let {an}(n goes from 1 to infinity) be a sequence. For each n define:(adsbygoogle = window.adsbygoogle || []).push({});

sn=Summation(j=1 to n) of aj

tn=Summation(j=1 to n) of the absolute value of aj.

Prove that if

{tn}(n goes from 1 to infinity)

is a Cauchy sequence, then so is

{sn}(n goes from 1 to infinity).

I started this proof with the definition of a Cauchy sequence. Pick an N large enough so that n,m>N makes

|an - am| < epsolon.

So if tn is Cauchy, we have

|tn-tm| < epsolon.

tn-tm = summation|an|-summation|am| = |an|+|an-1|+...+|am+1|

so now

|an| + |an-1| +...+ |am+1| < epsolon

but

|an + an-1 + ... + am+1| < |an|+|an-1|+...+|am+1|

by triangle inequality.

so now

|an + an-1 +...+ am+1| < epsolon

but

|an + an-1 + ... + am+1| = |sn - sm|

so now

|sn-sm| < epsolon, and therefore Cauchy.

Can anybody tell me if this makes sence? Or at least tell me how to write out "summation from n=1 to infinity" on here in symbols??? Thanks so much!

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# Homework Help: Cauchy sequences

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