Finite Dimensional Vector Space Over a Field is a Noetherian and Artinian $F$-module

Question

I'm trying to prove that if $V$ is a finite dimensional vector space over a field, $F$, then $V$ is a Noetherian and Artinian $F$-module.

I'm assuming I just have to prove that $V$ is Noetherian as an $F$-module, and then just switch all $\subseteq$ to $\supseteq$ for the Artinian proof.

I'm struggling as to how to get started with this.

Any suggestions?

Answer 1

Let be a increasing sequence of submodules( that is sub vector space) :

$V_1\subset V_2......$

But since V is finite dimensional vector space

$\bigcup V_i $

Is finite dimensional that is $\bigcup V_i =span\{v_1....v_n\}$

But since is a union there is for all $v_j$ an $V_{k_j}$ such that $v_j \in V_{k_j}$ let be $V_{max\{k_j\}}$ then

$\bigcup V_i=V_{max\{k_j\}}$

Therefore for all $n>max\{k_j\}$ $V_n=V_{max\{k_j\}}$

QED

For artinian is very similar