I was recently trying to prove something like
$$ \lim_{h\to 0} \int g(x, h) \,\text d\nu(x) $$ existed and had a certain value by using the dominated convergence theorem (DCT). But the DCT states that I need a sequence of function $f_n$ that converge pointwise almost everywhere to a function $f$, so this means that I need to somehow have $f_n$ with $n\to\infty$ represent this "continuous" limit of $h\to 0$.
I know my question is vague but mainly I want to know the following:
(1) is this a weird way to go about this in the first place? When I see something like $\lim _{h\to 0}$ I think of it as being "continuous" in the sense that it's like shrinking a ball around $0$ and looking at the image of that ball. I don't think of it as an indexed sequence of functions, but is it easy to do that? Can I look at something like the sup of the image with balls of radius $1/n$ or something like that?
(2) what are the general strategies for turning a non-sequence limit like this into a limit of a sequence of functions indexed by $\mathbb N$? What I've been exploring is something like considering an arbitrary sequence $a_n$ with $a_n \to 0$ and then letting $f_n(x) = g(x, a_n)$, but this feels weird.
Note that the limit of a function can be characterized by sequence:
$\lim_{x\rightarrow a}f(x)=L$ if and only if for every sequence $(a_{n})$ such that $a_{n}\ne a$ and $a_{n}\rightarrow a$, then $f(a_{n})\rightarrow L$.
And you are right to assume that $f_{n}(x)=g(x,a_{n})$, this is by the above sequential characterization.