proving a random variable is a martingale 
I am on the final part. I have shown all the properties of martingales, except for the fact that $E|N_n| < \infty$. The solutions state $|N_n|$ is bounded, but I don't see how it is as $S_n$ is not bounded.

$4S_n-n\leq 4n-n=3n$, and $4S_n-n\geq -4n-n=-5n$. Therefore $|N_n|\leq 5n$, so $\mathbb{E}[|N_n|]\leq 5n<\infty$.
You're right that the sequence of random variables $\{N_n\}$ is not uniformly bounded, but each individual random variable $N_n$ is bounded. 
