Questions about the proof of Theorem 13.4 in Matsumura‘s Commutative Ring Theory

I'm reading the Matsumura's Commutative Ring Theory and I'm trying to understand theorem 13.4.

Suppose $A$ is Noetherian semilocal ring, $M$ is a finite $A$-module, $\mathfrak{m}$ is the Jacobson radical of $A$. Define $d(M)$ is the degree of Samuel function $\chi_M(n)$ of $M$ and $\delta(M)$ is the samllest value of $n$ s.t. there exist $x_1,\cdots,x_n\in\mathfrak{m}$ for which $l(M/x_1M+\cdots+x_nM)<\infty$.

I have some questions about the proof in the step 3.:

Theroem 13.4 Let$ A $ be a semilocal Noetherian ring and $M$ a finite $A$-module; then we have$$\text{dim}\ M=d(M)=\delta(M)$$proof.Step 3. We show that $\text{dim}\ M\geq\delta(M)$, by introduction on $\text{dim}\ M$. If $\text{dim}\ M=0$ then $\text{Supp}(M)\subset\text{m-Spec}\ A=V(\mathfrak{m})$ so that for large enough $n$ we have $\mathfrak{m}^n\subset\text{ann}(M)$, and $l(M)<\infty$, therefore $\delta(M)=0$. Next suppose that $\text{dim}\ M>0$, and let $\mathfrak{p}_i$ for $1\leq i\leq t$ be the minimal prime divisors of $\text{ann}(M)$ with $\text{coht}\mathfrak{p}_i=\text{dim}\ M$; then the $\mathfrak{p}_i$ are not maximal ideals, so do not contain $\mathfrak{m}$. Hence we can choose $x_1\in\mathfrak{m}$ not contained in any $\mathfrak{p}_i$. Setting $M_1=M/x_1M$ we get $\text{dim}\ M_1<\text{dim}\ M$. Therefore by the inductive hypothesis $\delta(M_1)\leq\text{dim}\ M_1$; but obviously $\delta(M)\leq\delta(M_1)+1$, so that $\delta(M)\leq\text{dim}\ M_1+1\leq\text{dim}\ M$.

Question1: I don't know how to get $\mathfrak{m}^n\subset\text{ann}(M)$, and $l(M)<\infty$ by using $\text{Supp}(M)\subset\text{m-Spec}\ A=V(\mathfrak{m})$.

Question2: Choose $x_1\in\mathfrak{m}$ not contained in any $\mathfrak{p}_i$.Then Setting $M_1=M/x_1M$. How to get $\text{dim}\ M_1<\text{dim}\ M$ ?