Originally Posted by
Drexel28 You're thinking about this way too hard. What you want to prove is that any ascending chain terminates, right? But for any (non-empty) subsets $\displaystyle A_1,\cdots,A_{n+1}$ of your ring $\displaystyle R$ with $\displaystyle n$ elements if $\displaystyle A_1\subsetneq\cdots\subsetneq A_{n+1}$ then this implies that $\displaystyle \#(A_1)<\cdots\#(A_n)$ and so evidently $\displaystyle \#(A_{n+1})\geqslant n+\#(A_1)\geqslant n+1$...but this isn't possible since $\displaystyle \#(A_{n+1})\leqslant \#(R)=n$. Get the point?