If $\int_{t_1}^{t_2} f_n(t) dt$ has a limit as $n \to \infty$ for dense points $t_1, t_2 \in \mathbb{R}$, limit exists for all finite intervals?

Question

Let $\{ f_n \}$ be a sequence of non-negative locally-integrable functions on $\mathbb{R}$.

Further suppose that there exists a dense subset $Q \subset \mathbb{R}$ such that \begin{equation} \int_{t_1}^{t_2} f_n(t)dt \end{equation} converges to a finite limit as $n \to \infty$ for any pair $(t_1,t_2) \in Q^2$.

Then, my question is that for any $a,b \in \mathbb{R}$, does the sequence \begin{equation} \int_{a}^{b} f_n(t)dt \end{equation} converges to a finite limit as $n \to \infty$?

I suspect that this is true since $f_n$'s are assumed to be nonnegative, but cannot justify easily.. Could anyone please help me?