On keisler's book, Elementary Calculus, he presented the following definition:
$\epsilon\ \approx\ \delta$ compared to $\Delta x$ if $\epsilon/ \Delta x \approx\ \delta\ / \Delta x$
and used it to prove the infinite sum theorem
$B(a, b) = \int_a^b h(x)\ dx$ if $\Delta B\ \approx\ h(x) \Delta x $ compared $ \Delta x$
Here's the proof:
And he kept using it to find formulas for different quantities: area, volume, .. etc. but he never assures the correctness of the "$\Delta B\ \approx\ h(x) \Delta x$ compared $\Delta x$" part explicitly. Does he take that condition for granted as being intuitive in proving the formulas he derived thereafter? Is there a way to assure it that I am missing?