Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning the set of all consecutive integers from $a_n$ to $b_n$.. Let $A_{\alpha},\alpha \in \mathcal{A}$ be an uncountable family of infinite subsets of $\mathbb{N}$ that satisfies the following:
There exists some $c \in (0,1)$ so that for all $\alpha \in \mathcal{A}$ there exists some $n(\alpha) \in \mathbb{N}$ so that $|A_{\alpha} \cap I_n|\geq c(b_n-a_n)$ for all $n\geq n(\alpha)$. (Here $|B|$ is the number of elements of a finite subset $B \subset \mathbb{N}$)
Then my question is the following:
Does there exist a strictly increasing sequence of positive integers $\{l_n\} \to \infty$ for which there exists an infinite subset $\mathcal{B}\subset \mathcal{A}$ so that $l_n \in A_{\alpha}$ for all $n \in \mathbb{N}$, $\alpha \in \mathcal{B}$?