- #1
elliti123
- 19
- 0
Hello , i was just wondering if anyone could clarify one thing in this proof (its from Konrad Knopp book on infinite series) : If (x0,x1,...) is a null sequence, then the arithmetic means
xn'= x0+x1+x2+...+x/n+1 (n=1,2,3,...)
also forms a null sequence.
Proof: If ε >0 is given, then m can be so chosen, that for every n > m we have |xn| < ε/2 . For these n's, we have
|xn'| ≤ |x1+x2+x3+...+xm| / n+1 +(ε/2) (n-m /n+1)
since the numerator of the first fraction on the right hand side now contains a fixed number, we can further determine n0, so that for n > n0 that fraction remains < ε/2. But then, for every n > n0 , we have |xn'| < ε and our theorem is proved.
My question is: the chosen m in the proof as far as i know is a natural number changing according to what epsilon we give it so for example if the chosen m is 3 it might work for a particular ε but might not for another ε less than the other ε we have chosen first . So i have come to a conclusion that m or n0 that every n should be more than so the sequence converges to a real number is a function of epsilon therefore it changes whenever epsilon does. Now how exactly is the numerator they describe in the proof a fixed number?
Since the m changes whenever ε does then it is logical to infer that the summation of those terms would obviously change. And would you please explain the last part of the proof after the inequality i seem to have some vivid idea but i don't think i still get the last part. Thanks.
xn'= x0+x1+x2+...+x/n+1 (n=1,2,3,...)
also forms a null sequence.
Proof: If ε >0 is given, then m can be so chosen, that for every n > m we have |xn| < ε/2 . For these n's, we have
|xn'| ≤ |x1+x2+x3+...+xm| / n+1 +(ε/2) (n-m /n+1)
since the numerator of the first fraction on the right hand side now contains a fixed number, we can further determine n0, so that for n > n0 that fraction remains < ε/2. But then, for every n > n0 , we have |xn'| < ε and our theorem is proved.
My question is: the chosen m in the proof as far as i know is a natural number changing according to what epsilon we give it so for example if the chosen m is 3 it might work for a particular ε but might not for another ε less than the other ε we have chosen first . So i have come to a conclusion that m or n0 that every n should be more than so the sequence converges to a real number is a function of epsilon therefore it changes whenever epsilon does. Now how exactly is the numerator they describe in the proof a fixed number?
Since the m changes whenever ε does then it is logical to infer that the summation of those terms would obviously change. And would you please explain the last part of the proof after the inequality i seem to have some vivid idea but i don't think i still get the last part. Thanks.
Last edited:
