Let $\mathscr{O}$ be a discrete valuation ring with uniformizer $\pi$ and ring of formal power series $\mathscr{O} [[X]]$. I can understand untuitively "why" it is the case that the quotient $\mathscr{O} [[X]] / \langle X^t , \pi^r \rangle $ (by the ideal) is an $\mathscr{O}$-module of finite length for all $t,r \in \mathbb{N}$.
It seems more or less clear since it 'removes' all components consisting of higher powers, but I can't justify it formally, or by referencing other standard results. Perhaps I'm just missing something. Any help with this would be greatly appreciated.
Thanks,
M