Convergence in distribution; binomial example 
Hello, I hope everyone is doing well. I am currently having a hard time seeing where I am supposed to arrive at for this particular question. I will try my best to explain:
I know, conceptually, that this is a binomial distribution, and I need to show that the the random variable in standard form converges in distribution to the standard normal random variable $Z$. I understand that if enough samples are taken, it will slowly converge into a normal distribution. What I do not understand is how I would go about this. What exactly do I "show"? How would I do that? I cannot just say that i.i.d. random sample implies convergence in distribution; I must show it.
I also would like an explanation of how to go about part b). My instructor told me that I have to use the central limit theorem to show it converges to $0$ mean and $p(1-p)$ variance. I found this explicitly for the mean by doing
$$E[\sqrt{n}(\overline{X_n}-p)] = \sqrt{n}E[p-p] = 0 $$
He did not accept that as THE answer he wanted.
Any help would be great. I have been searching for examples for hours now. Here are some resources https://www.ma.imperial.ac.uk/~ayoung/m2s1/Convergencedistribution.PDF
https://www.youtube.com/watch?v=dRUm8U-r0IM
I cannot seem to wrap my head around doing this on paper. I know the answer is right in front of me, but I am stumped. I know this may seem like a trivial subject, but I am in a high paced statistics class. I do not have a lot of time to learn things thoroughly.
 A: The central limit theorem says if $X_1,X_2,\dots$ are independent observations from any distribution with mean $\mu$ and variance $\sigma^2$ then $\frac{\bar X_n-\mu}{\sigma/\sqrt n}\rightarrow^d \mathcal N(0, 1)$.
If $X\sim\text{Bernoulli}(p)$ then $EX=p$ and $\sigma^2=p(1-p)$ and so the CLT says that $\sqrt n\frac{\bar X_n-p}{\sqrt{p(1-p)}}\rightarrow ^d\mathcal N(0, 1)$.
For part b, use part a. By Slutkey's lemma, $\frac{\sqrt n(\bar X_n-p)}{\sqrt{p(1-p)}}\cdot \sqrt {p(1-p)}\rightarrow \mathcal N(0, 1)\sqrt {p(1-p)}=\mathcal N(0, p(1-p))$.
A: As @Jellyfish pointed out, and as the questions explicitly direct you, each item is a direct consequence of a fundamental theorem in statistics:

*

*CLT

*Slutsky

*CMT

Perhaps you should review the statements of each?

My instructor told me that I have to use the central limit theorem to show it converges to 0 mean and p(1−p) variance. I found this explicitly for the mean by doing...

What you showed next does not prove convergence in distribution. It is merely an application of linearity of expectation. To show convergence in distribution, you have to show convergence in the cdfs at every point for which the cdf is continuous. Of course, in this case you don't have to do any of this explicitly; you can just invoke CLT.
