A submartingale is a real-valued stochastic process $X=\{X_n\}$ adapted to a filtration $\{\mathcal{F}_n\}$ such that $$E[X_{n+1}\mid \mathcal{F}_n] \geq X_n.$$ For a supermartingale just reverse the inequality.
So I don't want someone to just give me the answer, but I'm having trouble just coming up with a submartingale at all. I've been using Durrett's book and there is a lack of examples for sure. Could someone point me in the right direction? Thank you so much!
Read Theorem 4.2.7 on Durrett's book: enter image description here
It emphasizes the increasing function. On the other hand, $f(x)=x^2$ is decreasing for $x<0$. Therefore we can just set $X_n = -1/n$ then it's submartingale, $X_n^2$ is increasing then it's supermartingale.
You can even play with some randomness on this set to make them real 'variables'.