At Most Countable Sets: Finite vs Countable Quick question:
I understand that finite sets are equivalent to $J_n$ for some n $\in$ N, and that countable sets are equivalent to N. Also, either of these is true if and only if an injective map f : A $\rightarrow$ N exists or if and only if a surjective map g : N $\rightarrow$ A exists.
Also, the proof that $J_n$ is NOT equivalent to N goes:
Define $J_n = \{1,2,\ldots,n\}$ and consider the mapping h : $J_n \rightarrow$ N. Observe that h(j)  $\le \sum_{i=1}^nh(i)$ for all $j \in J_n$, so it follows h is not onto. Hence, $J_n \subset$ N.

Is it valid to claim $\bigcup_{A \in F}(A) \sim N$ can be shown, when A is finite for all A $\in$ F and F is countable, by recursively declaring the bijection:
k : $\bigcup_{i=1}^m(A) \rightarrow$ $J_n$ until m and n diverge toward infinity.

This would imply that as n gets large enough through summation, $J_n \sim N$ if and only if the $\bigcup_{i=1}^\infty(A)$ is countable.
 A: $A$ is a countable set iff there exists an injection $f: A \rightarrow \mathbb N$. So all finite sets are countable; but not all countable sets are finite.
Your attempt at proving that $J_n$ is "NOT equivalent to" $\mathbb N$ has a flaw. You correctly note that $J_n$ is a proper subset of $\mathbb N$. (Of course, we could just as easily prove this via: $n+1 \notin J_n$ and $n+1 \in \mathbb N$, therefore $J_n$ is a proper subset of $\mathbb N$).
But so what? Let $E$ be the set of even naturals. It's obvious that $E$ is a proper subset of $\mathbb N$; but that doesn't mean we can conclude that $E$ is a finite set!
To prove that a set is finite depends on what your definition of 'finite' is. The usual definition is that for some $n \in \mathbb N$, there exists a bijection $f : A \rightarrow J_n$; where $J_n = \{i \in \mathbb N: i \leq n\}$ and by that definition, the trivial function $f(x) = x$ shows that $J_n$ must always be finite; we don't need anything fancier than that.
As to your claim, your assertion is ill-defined. 
What do you mean, for an arbitrary $A \in F$ where $F$ is a countable set of finite sets, by $\bigcup_{i=1}^m(A)$? If $A$ is the set $\{7\}$ and $m=3$, what set are we unioning when $i=2$? Is $m$ supposed to depend of $A$ in some way, or is $A$ supposed to depend of $i$ or $m$ in some way?
When you have answered that, what exactly do you mean by "until $m$ and $n$ diverge to infinity"? I presume that $m$ and $n$ are both elements of $\mathbb N$; and I note that $\mathbb N \notin \mathbb N$. How do the words "infinity" and "diverge" fit into this picture?
To help focus your enquiry, the thing I might suggest is this:
Let $F$ be a countable set of finite sets $\{A_i\}$. 
To keep from being sloppy, I will first note that since $F$ is countable, there is an injective function $q: F \rightarrow \mathbb N$. Now $q(F)$ is a subset of the naturals, and therefore there is some unique $a \in F$ such that $q(a)$ is minimal in $q(F)$ (assuming that every set of naturals has a unique least element... obvious of course, but must previously be proven!!). 
So call that minimal $a$ to be $A_1$; now $q(F) - \{A_1\}$ is either empty, or it has a minimal element which we will call $A_2$. Et cetera! So whether $F$ is finite or not, I can give some meaning to $A_i \in F$.
Now, your claim seems to want to be: if $F = \{A_i\}$ is a countable collection of finite sets, and for all $n \in \mathbb N$, there exists $m \in \mathbb N$ such that there exists a surjection $f : \bigcup_{i=1}^m A_i \rightarrow J_n$, then exists a bijection $g: F \rightarrow \mathbb N$.
And that, my friend, can be proven!
