In this proof I know since R is noetherian it can be written as descending sequence of ideals which stabilizes after finite steps. Also I know since R is noetherian implies every ideal is finitely generated then I have some doubt;
1) Why $\cap \mathfrak m^n$ is an ideal of R and why $\mathfrak m\cdot \cap \mathfrak m^n=\cap \mathfrak m^n$ is unclear. Please help regarding this.
It should be clear that the intersection of ideals is an ideal, so that $\mathfrak{a}=\bigcap_n\mathfrak{m}^n$ is an ideal.
It really isn't obvious that $\mathfrak m\mathfrak a=\mathfrak a$. One way to prove this is to invoke the Artin-Rees lemma. A special case of this is that for any ideal $\mathfrak b$ then $\mathfrak m^{n+1}\cap\mathfrak b =\mathfrak m(\mathfrak m^n\cap\mathfrak b)$ for all large enough $n$. Taking $\mathfrak b=\mathfrak a$ gives $\mathfrak a=\mathfrak m\mathfrak a$.