Find finite structures $A_i$, $i\in I$ and an ultrafilter $U$ on $I$ so that the ultraproduct $\overset{}{\underset{i\in I}{\prod}} A_i/U$ is infinite. Hint: Take some really "simple" finite models $A_i$ that increases in size (say that $A_i$ is of size $i$) and a non-principal ultrafilter.
I am completely confused as to what to do. On my own, I can do nothing more than reciting definitions from the textbook. However, I saw that such a proposition exists:
Let $L$ be a countable language, $(M_i:i\in \Bbb N)$ a sequence of $L$-structures and $U$ a non-principal ultrafilter on $\Bbb N$. Then the ultraproduct $\overset{}{\underset{i\in \Bbb N}{\prod}} M_i/U$ is $\aleph_1$-saturated.
Could anyone let me know if proving this proposition is the right direction to go?
(I apologise for the poor quality of this question; I know I am supposed to have at least given it a try, but I am so lost that I can do nothing but recite mindlessly the definitions in the book. I have never done any serious maths in university until now, so I am really desperate)