I would like to compute the limit of CDF for a Binomial distribution as $n \rightarrow \infty$, \begin{equation*} \lim_{n \rightarrow \infty} F( \theta;n,q) = \lim_{n \rightarrow \infty }\sum_{k=0}^{\lfloor \theta n \rfloor} \binom{n}{k} q^k (1-q)^{n-k} \end{equation*}
An hint for this problem can be obtained in this link : The asymptotic behavior of the CDF of Binomial distribution $$ \lim_{n\to\infty}F(\theta;n,q) = \begin{cases}0 & \text{if } \theta < q\\1 & \text{if } \theta > q\end{cases}. $$ My question 1 is : Where and how to apply the Weak Law of Large Number to obtain the solution?
Further more, let $$G_n(\theta) = \sum_{k=0}^{\lfloor \theta n \rfloor} \binom{n}{k} \int_0^1 q^k (1-q)^{n-k} d W(q). $$
My Question 2 is how to prove that, $$ G_{\infty}(\theta) = W(\theta) \text{ in distribution.}$$