Why does the Riemann sum of a double integral use $\Delta x, \Delta y \rightarrow 0$ as its limit instead of $m,n \rightarrow \infty $?

Question

Edit: Let $m$ and $n$ be the number of subintervals that a region $R$ has been divided up into.

$\Delta x, \Delta y \rightarrow 0$ is equivalent to $m,n \rightarrow \infty $ because both of those limits just describe you taking smaller and smaller subrectangles of a region $R$; you take arbitrarily small subrectangles that are so small that they "approximate" the area perfectly.

So is there any reason that the Riemann sum of a double integral uses $\Delta x, \Delta y \rightarrow 0$ as its limit instead of $m,n \rightarrow \infty $ ? Feels like I'm missing something.

Answer 1

Even in $\Bbb R$ you can have partitions with an arbitrarily great number of points but with some subinterval of constant width $> 0$. Example in $[0,1]$: $$x_0 = 0, x_1 = 1/2, x_n = 1/2 + n/(2N), 2\le n\le N.$$