Specifically I mean:
What are some examples of when $\epsilon$, $\delta$ inequalities are used without implicitly trying to say that the limit of some net is approaching a certain value? When are inequalities used in analysis in a way that doesn't imply topology via convergence of nets?
For example, with inner regularity of measures, one is basically just relating a net (or directed set) of compact sets converging to our given measurable set with the net of real numbers corresponding to the measures of these compact sets -- inner regularity just says that the convergence of the first net implies the convergence of the second net and that the resulting limit of the second net also happens to be the measure of the measurable set being approximated from within.
In other words, all of the examples of $\epsilon, \delta$ inequalities, which I can think of right now, just consist of mapping nets (or directed sets) in some more abstract space to nets in the Euclidean topology of the real numbers -- indeed, this seems like this would characterize nets in any Hausdorff metric space.
I was trying to think of how I would explain analysis to middle schoolers, and my first thought was that analysi is the study of inequalities, but after thinking about it more, the only examples of inequalities in analysis I could think of were ways to obliquely refer to the concept of net in the Euclidean topology -- leading me to the conclusion that nets might actually be the more fundamental object of study in analysis.
However, there is probably a simple counterexample which would disavow me of this belief (that nets are more fundamental to analysis than inequalities), which is essentially what I am asking/looking for.