I'm having some trouble understanding what the following definition is saying, i think i have a basic understanding but would appreciate some clarification on the technical aspects. The precise meaning of the limit that defines the integral follows as:

for every number $\epsilon$ there is an integer N such that

$$\vert \; \int_a ^b f(x)dx - \sum_{i=1}^nf(x_i^*)\Delta x \;\vert < \epsilon $$

for every integer $n > N$ and for every choice of $x_i^*$ in $[x_{i-1}, x_i] $

I know that $\int_a^bf(x)dx$ is the integrand and that $ \sum_{i=1}^nf(x_i^*)\Delta x$ is a Riemann Sum (Approximation of the area under a curve by rectangles). Judging by the absolute value sign I would take this precise definition to be formalizing the idea of "closeness", or saying that the difference between the actual area, the integrand, and the approximation, the Riemann sum is extremely small (i.e highly accurate).

I don't understand how the integer N comes into it, in particular the last sentence "for every integer $n > N$ and for every choice of $x_i^*$ in $[x_{i-1}, x_i]$. I'd appreciate any clarification on this and confirmation of whether i have the basic idea of what this is saying, i unfortunately couldn't find any explanation of this elsewhere.


$N$ comes from the step $h_n$ of a regular ( equidistant) partition of the interval $[a,b]$ containing $n$ sub-intervals, defined by


to say that $h_n$ becomes small means $n$ is greater than a certain $N$ or

$$\lim_{h_n \to 0} \equiv \lim_{n\to +\infty}.$$


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