Let $f$ be a function that is Lebesgue integrable (w.r.t. the Borel measure $\lambda$) over $(0, \infty)$. Let $a, x > 0$ with $a < x$ and let $(a_n)_{n \in \mathbb{N}}$ be an increasing sequence of positive numbers converging to $a$.
Under what conditions does the equality $$ \lim_{n \to \infty} \int_{(a_n, x)} f(t) \text{d}\lambda(t) = \int_{(a, x)} f(t) \text{d}\lambda(t) $$ hold?
I know a lot of theorems about limits coupled with integrals, but only if the limit-variable ($n$ in our case) appears in the integrand, but here, it appears in the set over which we integrate (the boundaries of integration).
Note that $f$ is not (necessarily) Riemann integrable, because in that case, the Riemann and Lebesgue integral of $f$ (over compact sets) coincide, and if the integrals in the expression were Riemann integrals, we know that the equality holds.
$\lvert f\cdot I_{(a_n,x)}\rvert\le\lvert f\cdot I_{(0,\infty)}\rvert$ and $\int \lvert f\cdot I_{(0,\infty)}\rvert<\infty$ (since you specified that $f$ is integrable over $(0,\infty)$), and your result follows from the Lebesgue dominated convergence theorem.