Definition for series with negative index and order of taking limits

Question

I have thee questions and they seem all related to me and every number i say is complex number below.

My definition for $\sum_{k=0}^n=a_k$ is by induction. That is, given finite set of numbers $\{a_0,...,a_n\}$ with $a_i=0 (i>n)$, $\sum_{k=0}^{m+1}a_k=\sum_{k=0}^{m}a_k+a_{m+1}$.

It may seem nonsense to define finite sums 'precisely', but i really like this definition since it's abstract which, i believe, is more suitable to mathematics.

I also want to know if there is another precise definition for sums, since this definition is hard to handle objects sometimes.

1(Done).What is the precise definition for $\sum_{i\in I} a_i$ when $I$ is finite?

2(Done). What is the precise definition for $\sum_{-N}^N c_n$?

1. Let $\alpha$ be a monotonically increasing function an $f$ be a Riemann-Stieltjes Integrable function on every connected compact set in $(a,b)$. (My definition for Riemann-Stieltjes Integeal is written below). How do i prove that $\lim_{x\to a}\lim_{y\to b} \int_{x}^{y} f d\alpha = \lim_{y\to b}\lim_{x\to a} \int_{x}^{y} f d\alpha$? ($a,b$ may also be $\infty,-\infty$.)

*Riemann-Stieltjes Integral;

Let $f$ be a function defined on $[a,b]$. Then $f\in\mathscr{R}$ iff $\inf_P U(P,f,\alpha)=\sup_P L(P,f,\alpha)$ ($P$ is a partition of $[a,b]$.

EDIT; I have proved that "If $I\approx n^+$ is a subset of a field and $f&g$ are bijections from $n^+$ to $I$, then $\sum_{k=0}^n f(k)=\sum_{k=0}^n g(k)$. So 1,2 are done, but 3 is still in question..

Answer 1

The third question, on the Riemann-Stiltjes integral, does not appear to have much to do with the other two. It should have been posted separately. I would approach it this way: fix a point $c\in (a,b)$. Since $x\to a$ and $y\to b$, it suffices to consider the values of $x,y$ such that $a<x<c<y<b$. The integral is additive with respect to the interval of integration $\int_x^y f\,d\alpha = \int_x^c f\,d\alpha +\int_c^y f\,d\alpha $. Observe that one of the summands depends only on $x$ while the other depends only on $y$. It is a general fact that for any two complex-valued functions $F,G$ $$\lim_{x\to a}\lim_{y\to b} (F(x)+G(y)) = \lim_{y\to b}\lim_{x\to a} (F(x)+G(y))$$ because both limits are simply $\lim_{x\to a} F(x)+ \lim_{y\to b} G(y)$. To be more precise, the following are equivalent:

1. $\lim_{x\to a}\lim_{y\to b} (F(x)+G(y))$ and $\lim_{y\to b}\lim_{x\to a} (F(x)+G(y))$ exist;
2. $\lim_{x\to a}\lim_{y\to b} (F(x)+G(y))$ and $\lim_{y\to b}\lim_{x\to a} (F(x)+G(y))$ exist and are equal;
3. $\lim_{x\to a} F(x)$ and $ \lim_{y\to b} G(y)$ exist

Suggested proof strategy: $1\implies 3\implies 2\implies 1$