Real Analysis I need to prove the question below. In addition, I need to have a detailed explanation of each step. Question: If the Series an from n=1 to infinity is a convergent series of positive numbers and if the sequence {ani} from i=1 to infinity is a subsequence of {an} from n=1 to infinity, prove that the series ani from i=1 to infinity converges. This is what I have so far. Let {Snk} be the sequence of partial sums for the series ani from i=1 to infinity. Since all elements of an are positive numbers then ani >0 for all i. Therefore {Snk} is increasing. (Question - can I just state this or do I need to prove it? My professor wants very exact proofs) As the series an from n=1 to infinity is convergent, then by definition {Sn} the sequence of partial sums is convergent to some number, say L. So then an arbitrary element of {Snk} , the finite sum of ani from i=1 to k will be less than or equal to an arbitrary element of {Sn}, the finite sum of ai from i=1 to nk which is less than or equal to L. As this is true for arbitrary elements of {Snk} and {Sn} then {Snk} <=L for all k in N the natural numbers so Snk is bounded above. Thus Snk is increasing and bounded above and Snk is convergent and by definition the series ani from i=1 to infinity converges. Any flaws in this reasoning? Have I stated this clearly?
I need to prove the question below. In addition, I need to have a detailed explanation of each step.
Question:
If the Series an from n=1 to infinity is a convergent series of positive numbers and if the sequence {ani} from i=1 to infinity is a subsequence of {an} from n=1 to infinity, prove that the series ani from i=1 to infinity converges.
This is what I have so far.
Let {Snk} be the sequence of partial sums for the series ani from i=1 to infinity. Since all elements of an are positive numbers then ani >0 for all i. Therefore {Snk} is increasing. (Question - can I just state this or do I need to prove it? My professor wants very exact proofs)
As the series an from n=1 to infinity is convergent, then by definition {Sn} the sequence of partial sums is convergent to some number, say L.
So then an arbitrary element of {Snk} , the finite sum of ani from i=1 to k will be less than or equal to an arbitrary element of {Sn}, the finite sum of ai from i=1 to nk which is less than or equal to L.
As this is true for arbitrary elements of {Snk} and {Sn} then {Snk} <=L for all k in N the natural numbers so Snk is bounded above.
Thus Snk is increasing and bounded above and Snk is convergent and by definition the series ani from i=1 to infinity converges.
Any flaws in this reasoning? Have I stated this clearly?
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