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 a n from n=1 to infinity is a convergent series of positive...

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 a_n
from n=1 to infinity is a convergent series of positive numbers and if the sequence {an_i} from i=1 to infinity is a subsequence of {a_n} from n=1 to infinity, prove that the series an_i
from i=1 to infinity converges.

This is what I have so far.

Let {Sn_k} be the sequence of partial sums for the series an_ifrom i=1 to infinity. Since all elements of a_n
are positive numbers then an_i
>0 for all i. Therefore {Sn_k} is increasing. (Question - can I just state this or do I need to prove it? My professor wants very exact proofs)

As the series a_n
from n=1 to infinity is convergent, then by definition {S_n} the sequence of partial sums is convergent to some number, say L.

So then an arbitrary element of {Sn_k} , the finite sum of an_i
from i=1 to k  will be less than or equal to an arbitrary element of {S_n}, the finite sum of a_i
from i=1 to n_k
which is less than or equal to L.

As this is true for arbitrary elements of {Sn_k} and {S_n} then {Sn_k} <=l for="" all="" k="" in="" n="" the="" natural="" numbers="" so="">_k
is bounded above.

Thus Sn_k
is increasing and bounded above and Sn_k
is convergent and by definition the series an_i
from i=1 to infinity converges.

Any flaws in this reasoning? Have I stated this clearly?