Law of large numbers and cardinality of intersection

Question

Let sets $A,B$ s.t. $A\subseteq B$ with cardinality $|A|=\frac{q}{n}$ for integers $q,n$ and $q$ a multiple of $n$. Assume that the sample space is finite and let's consider an element $x$. In the context of some problem it has been shown that $$\mathrm{Pr}(x\in B)=\frac{m}{n}p^{m-1}(1-p)^{n-m}$$ for some fixed $m\in\mathbb{N}^{+}$ and some fixed $p\in[0,1)$. At this point it is stated that as $q$ gets large (and $n$ remains sublinear in $q$), by the law of large numbers $$|A\cap B|=|A|\cdot\mathrm{Pr}(x\in B)+o(q)$$ where $o(\dots)$ denotes the small-oh notation i.e. for functions $f(x)$, $g(x)$, $f(x)=o(g(x))$ if $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0$. I cannot understand the last derivation based on the law of large numbers and why we need to consider $q\rightarrow\infty$.