Let $a, b, p \in (0, 1)$. What is the distribution of the sum of $n$ independent Bernoulli random variables with parameter $p$? By considering this sum and applying the weak law of large numbers, identify the limit
$$ \lim_{n \to \infty} \sum_{r \in \mathbb{N}:an<r<bn} \binom{n}{r} p^r(1 − p)^{n−r} $$ in the cases (i) $p < a$; (ii) $a < p < b$; (iii) $b < p$.
I feel like the answers should be (i) $0$, (ii) $1$, (iii) $0$, but I don't know how to answer this question rigorously.
WLLN says that the probability that the sample mean of $n$ Bernoulli variables deviates from $p$ by more than any $\epsilon > 0$ goes to zero as $n \to \infty$. This means the probability that a binomial random variable deviates from $np$ by more than $n \epsilon$ goes to zero as $n \to \infty$.
Now take $0<\epsilon<\min \{ |p-a|,|p-b| \}$. Then in case (i) and (iii), $(an,bn)$ doesn't intersect with $((p-\epsilon)n,(p+\epsilon)n)$, thus it is contained in the complement, so
$$P(X \in (an,bn)) \leq P(X \not \in ((p-\epsilon)n,(p+\epsilon)n)) \to 0.$$
In case (ii), instead $(an,bn)$ contains all of $((p-\epsilon)n,(p+\epsilon)n)$ so you have
$$P(X \in (an,bn)) \geq P(X \in ((p-\epsilon)n,(p+\epsilon)n)) \to 1.$$
It may help to draw a number line to help see the set relationships going on here.