Let $M=\cdots \oplus M_{-1} \oplus M_0 \oplus M_1 \oplus \cdots$ be a finitely generated graded $k[x_1, \dots, x_n]$-module where $\deg x_i = 1$ for all $i$.
I would like to show that each component $M_s$ is a finite dimensional vector space over $k$ using the fact:
If $R$ is Noetherian and $M$ is a finitely generated $R$-module, then $M$ is Noetherian.
My attempt:
Suppose some $M_s$ is not finite dimensional over $k$.
Then $M_s \oplus M_{s+1} \oplus \cdots$ is not finite dimensional over $k$.
Since $M$ is Noetherian as a $k[x_1, \dots, x_n]$-module, we know $M_s \oplus M_{s+1} \oplus \cdots$ is finitely generated over $k[x_1, \dots, x_n]$.
Not sure where to go from here.