Localization of infinite variable ring over a prime ideal is not Noetherian

Question

$k$ is a field and consider the infinite variable polynomial ring $k[x_1,...,x_n,....]$. Let's take the prime ideal generated by $x_1,x_2,...,x_n$, i.e $P=(x_1,...,x_n)$. Next, look at the localization at $P$, $k[x_1,...,x_n,x_{n+1},....]_{P}$. I want to show, this ring is Noetherian. I was trying to solve the Nagata's example of constructing a Noetherian ring of infinite dimension. And this appears as an intermediate step. Can someone please help me? Thanks.