In the book "Local cohomology : An algebraic introduction with geometric application", page 289 there is a proof of the following theorem :

Assume that $R=\bigoplus_{n}R_{n}$ is positive graded and homogeneous, and let $M=\bigoplus_{n}M_{n}$ be a non-zero finitely generated graded $R$-module. Then $M$ can be generated by homogeneous elements of degrees not exceeding $\text{reg}M$.

Here is a part of the proof that I concern :

Let $N$ be the graded submodule of $M$ generated by $\bigoplus_{\le\text{reg}M}M_{n}$. It is suffices to show that $M_{\mathfrak{p}_{0}}=N_{\mathfrak{p}_{0}}$ for each $\mathfrak{p}_0$ in SpecR.....(argument).....( $M_{\mathfrak{p}_{0}}=N_{\mathfrak{p}_{0}}$ is proven). Then it is therefore enough to establish the claim in the statement of the theorem under the additional assumption that $R_0$ is local.

My question is :

1- We can't conclude that $M=N$ if we have $M_{\mathfrak{p}_{0}}$=$N_{\mathfrak{p}_{0}}$ ?

2- Why can we only consider only the case that $R_0$ is local? What is the idea behind that localization technique ?