Let $K[[x]]$ be the ring of formal power series over a field $K$, and $K[[x]] \supset (x) \supset (x^2) \supset (x^3 )\supset\ldots\supset(x^n )\supset\ldots$ be uniserial ring (that is, all its ideals are linearly ordered with respect to inclusion).
Prove that the rings $K[[x]]/(x^n)$ are artinian for every $n\geq 1$.
My reasoning is that $K[[x]]/(x^n)$ is of finite dimension. (That is, it has a finite basis). Secondly, the ideals will terminate eventually. That means it is artinian. However, I have failed to construct the proof in a better way.
From what you have written the ring has finitely many ideals. It has $n+1$ to be exact.
From there it is a no-brainer that all chains are eventually constant.