Definition. Let $F$ be a function of two variables that is defined in
some circular region around $(a,b)$. The standard limit of $F$ as $(x,y)$
approaches $(a,b)$ equals $L$ if and only if for every $\epsilon > 0$ there
exists $\delta > 0$ such that $F$ satisfies:
$$
| F(x,y) - L | < \epsilon
$$
whenever the distance between $(x,y)$ and $(a,b)$ satisfies:
$$
0 < \sqrt{ (x-a)^2 + (y-b)^2 } < \delta
$$
We will use the following notation for such limits of functions of two
variables:
$$
\lim_{(x,y)\rightarrow (a,b)} F(x,y) = L
$$
If we replace $(a,b)$ by infinity, then for two dimensions the definition would
become slightly different, namely:
$$
\lim_{|(x,y)|\rightarrow \infty} F(x,y) = L
$$
If and only if for every $\epsilon > 0$ there exists $N > 0$ such that $F$
satisfies:
$$
| F(x,y) - L | < \epsilon
$$
Whenever:
$$
\sqrt{ x^2 + y^2 } > N
$$
However, this case doesn't deserve much attention, because we can always write
the following:
$$
\lim_{|(x,y)|\rightarrow \infty} F(x,y) =
\lim_{(x,y)\rightarrow (0,0)} F(1/x,1/y)
$$
Thus reducing the $\infty$ limit to 2-D standard.
Therefore we consider the following iterated limit:
$$
\lim_{y\rightarrow b} \left[ \lim_{x\rightarrow a} F(x,y) \right] = L
$$
Theorem.
$$
\lim_{y\rightarrow b} \left[ \lim_{x\rightarrow a} F(x,y) \right] =
\lim_{x\rightarrow a} \left[ \lim_{y\rightarrow b} F(x,y) \right] =
\lim_{(x,y)\rightarrow (a,b)} F(x,y)
$$
Proof. We split the iterated limit in two pieces:
$$
\lim_{x\rightarrow a} F(x,y) = F_a(y)
$$
And:
$$
\lim_{y\rightarrow b} F_a(y) = L
$$
Thus it becomes evident that the iterated limit is actually defined as
follows.
For every number $\epsilon_x > 0$ there is some number $\delta_x > 0$ such that:
$$
| F(x,y) - F_a(y) | < \epsilon_x \quad \mbox{whenever}
\quad 0 < | x - a | < \delta_x
$$
For every number $\epsilon_y > 0$ there is some number $\delta_y > 0$ such that:
$$
| F_a(y) - L | < \epsilon_y \quad \mbox{whenever}
\quad 0 < | y - b | < \delta_y
$$
Applying the triangle inequality $|a| + |b| \ge |a+b|$ gives:
$$
| F(x,y) - F_a(y) | + | F_a(y) - L | \ge | F(x,y) - L | \quad \Longrightarrow \quad
$$ $$
| F(x,y) - L | < \epsilon_x + \epsilon_y
$$
On the other hand we have:
$$
0 < | x - a | < \delta_x \quad \mbox{and} \quad 0 < | y - b | < \delta_y \quad \Longrightarrow \quad
$$ $$
0 < \sqrt{ (x-a)^2 + (y-b)^2 } < \sqrt{ \delta_x^2 + \delta_y^2 }
$$
This is exactly the definition of the above standard limit of a function of two
variables if we only put:
$$
\epsilon = \epsilon_y + \epsilon_y \quad \mbox{and} \quad \delta = \sqrt{\delta_x^2 + \delta_y^2}
$$
Therefore:
$$
\lim_{y\rightarrow b} \left[ \lim_{x\rightarrow a} F(x,y) \right] =
\lim_{(x,y)\rightarrow (a,b)} F(x,y)
$$
In very much the same way we can prove that:
$$
\lim_{x\rightarrow a} \left[ \lim_{y\rightarrow b} F(x,y) \right] =
\lim_{(x,y)\rightarrow (a,b)} F(x,y)
$$
Consequently:
Iterated limits do always commute.
Corollaries.
1.
Iterated limits simply turn out to be special cases of the standard limit in
two dimensions.
2.
If the standard limit does exist, then associated iterated limits do also exist.
If the standard limit does not exist, then associated iterated limits do not
exist as well.
3.
Iterated limits always commute, if they exist. Iterated limits that "do not
commute" do not exist. But the reverse is not true. Iterated limits that
"commute" (by coincidence) do not necessarily exist. Mind the quotes.
4.
The proof is quite standard, surprisingly simple and it doesn't make
explicit use of the Axiom that Completed infinity is not given.
5.
As far as limits are concerned, there is no need, apparently, for
introducing a separate Axiom. The reason is that such an Axiom
is already part of the limit definition itself.
6.
It is tempting to even say that the Axiom can safely be restated as:
Infinities can only be approached through limits and nothing else.
7.
If the Axiom is relevant to science, rather than to
mathematics, then the apparent commutativity of iterated limits, in scientific
applications, may have tremendous consequences for the more general question
what mathematics is relevant for science and what not.