Problem: Give an example of submartingale $\{X_n\}$ with $\sup_nE |X_{n-1}-X_n|<\infty$ and stopping time $N$ with $E[N]<\infty$ such that $\{X_{n\wedge N}\}$ is not uniformly integrable.
Attempt: I think this asks us to give a counterexample to the optional stopping theorem with bounded increments. Since we have a finite a.s. stopping time condition, I think the example maybe some kind of low dimensional random walk. But I am not really sure..
Are you sure you have the assumptions correct? This link Here has a Theorem 4.7.5 which states that if $\{X_n\}_{n\geq0}$ is a sub-martingale and $\mathbb{E}\big[|X_{n+1} - X_n|\; \big| \; \mathcal{F}_n\big] < B < \infty$ and $\mathbb{E}[N] < \infty$ then $\{X_{n \wedge N}\}_{n \geq 0}$ is uniformly integrable.
Taking expectations $\mathbb{E}\Big[\mathbb{E}\big[|X_{n+1} - X_n| \; \big| \; \mathcal{F}_n\big]\Big] = \mathbb{E}\big[|X_{n+1} - X_n|\big] < B$. Therefore, it must be that $\{X_n\}_{n\geq1}$ cannot have almost surely uniformly bounded increments for your result to hold.
If you have uniformly bounded increments, then the Theorem suggests that such an example by proof is not possible.