Dominated convergence theorem - what sequence?

The statement of the dominated convergence theorem (DCT) is as follows:

"Sequential" DCT. Suppose $\{f_n\}_{n=1}^\infty$ is a sequence of (measurable) functions such that $|f_n| \le g$ for some integrable function $g$ and all $n$, and $\lim_{n\to\infty}f_n = f$ pointwise almost everywhere. Then, $f$ is an integrable function and $\int |f-f_n| \to 0$. In particular, $\lim_{n\to\infty}\int f_n = \int f$ (by the triangle inequality). This can be written as $$ \lim_{n\to\infty}\int f_n = \int \lim_{n\to\infty} f_n.$$

(The statement and conclusion of the monotone convergence theorem are similar, but it has a somewhat different set of hypotheses.)

As you note, the statements of these theorems involve sequences of functions, i.e., a $1$-discrete-parameter family of functions $\{f_n\}_{n=1}^\infty$. To apply these theorems to a $1$-continuous-parameter family of functions, say $\{f_\epsilon\}_{0<\epsilon<\epsilon_0}$, one typically uses a characterization of limits involving a continuous parameter in terms of sequences:

Proposition. If $f$ is a function, then $$\lim_{\epsilon\to0^+}f(\epsilon) = L \iff \lim_{n\to\infty}f(a_n) = L\quad \text{for $\mathbf{all}$ sequences $a_n\to 0^+$.}$$

With this characterization, we can formulate a version of the dominated convergence theorem involving continuous-parameter families of functions (note that I use quotations to title these versions of the DCT because these names are not standard as far as I know):

"Continuous" DCT. Suppose $\{f_\epsilon\}_{0<\epsilon<\epsilon_0}$ is a $1$-continuous-parameter family of (measurable) functions such that $|f_\epsilon| \le g$ for some integrable function $g$ and all $0<\epsilon<\epsilon_0$, and $\lim_{\epsilon\to0^+}f_\epsilon=f$ pointwise almost everywhere. Then, $f$ is an integrable function and $\int |f-f_\epsilon|\to 0$ as $\epsilon\to 0^+$. In particular, $$ \lim_{\epsilon\to0^+}\int f_\epsilon = \int \lim_{\epsilon\to0^+} f_\epsilon.$$

The way we use the continuous DCT in practice is by picking an arbitrary sequence $\pmb{a_n\to 0^+}$ and showing that the hypotheses of the "sequential" DCT are satisfied for this arbitrary sequence $a_n$, using only the assumption that $a_n\to 0^+$ and properties of the family $\{f_\epsilon\}$ that are known to us.

Let's look at it in a sample case. We want to prove by DCT that $$\lim_{\varepsilon\to0^+} \int_0^\infty e^{-y/\varepsilon}\,dy=0$$

This is the case if and only if for all sequences $\varepsilon_n\to 0^+$ it holds $$\lim_{n\to\infty}\int_0^\infty e^{-y/\varepsilon_n}\,dy=0$$

And now you can use DCT on each of these sequences. Of course, the limiting function will always be the zero function and you may consider the dominating function $e^{-x}$.