In my calculus notes, my professor presented the following proposition:
Proposition: Let $\sum_{n=1}^{\infty}a_n$ be a convergent series and let $1=n_1<n_2<n_3<…$ be an increasing sequence of indices. Consider the set of grouped sums of the sequence between these indices, i.e; $$A_k=\sum_{n=n_k}^{n_{k+1}-1}a_n.$$ Then, $$\sum_{k=1}^{\infty}A_k=\sum_{n=1}^{\infty}a_n$$
This proposition is supposed to say that if an infinite series converges, then the associative property holds. I was reading an equivalent proposition, but written in a different way, and I was able to prove it. However, I can't prove the statement the way my professor wrote it. Any help is welcome
Addendum:
The proposition I was able to prove is Exercise 2.5.3 (a) on page 65 from the book Understanding Analysis, 2nd ed by Stephen Abbott.
Best Answer
Hint: $$\sum_{k=1}^N A_k =\sum_{i=1}^{n_{N+1}-1} a_i$$ because finite sums are associative.