Using the weak law of large numbers to find the limit of $\sum\limits_{r = an}^{bn} {n \choose r } p^r (1-p)^{n-r}$

Question

I need to find the limit of $\sum\limits_{r}^{} {n \choose r } p^r (1-p)^{n-r}$ such that $ an < r < bn $ in the cases $p < a$, $ a < p < b$, and $b < p$.

I know I need to consider the sum of $n$ identically distributed independent Bernoulli random variables, but I'm not sure how to apply the weak law to show what is required. Any help you could offer would be very much appreciated.

Answer 1

Let $X_n$ be binomial with parameters $p$ and $n$. The desired sum is $P(a<X_n/n<b)$. But by the WLLN, $X_n/n\xrightarrow{p} p$, so ...