Let $Y_n$ be binomial distributed with parameters $n = n$ and $p = \frac{3}{4}$, show $\lim_{n \to \infty} \mathbb{P}(4Y_n \leq 3n) = \frac{1}{2}$

Question

So as the title says, we have random variable $Y_n$ which is binomial distributed with parameters $n = n$ and $p = \frac{3}{4}$, show $\lim_{n \to \infty} \mathbb{P}(4Y_n \leq 3n) = \frac{1}{2}$.

My attempt isn't good enough to write something about, just random notes. I figured the central limit theorem would be a way to solve the problem but I am currently clueless.

Answer 1

$$\mathbb P\left(4Y_n \le 3n\right) = \mathbb P\left(Y_n - \frac34 n \le 0\right) = \mathbb P\left(\frac{Y_n - \frac34 n}{\sqrt{n\times \frac34 \left(1 - \frac34\right)}}\le 0\right) \to \mathbb P\left(\mathcal N(0,1) \le 0\right) = \frac12$$