I am self learning Commutative Algebra, now I've started studying Completion of a ring and a module, I'm following the book Commutative Algebra by N S Gopalakrishnan. What I've got from this book is Suppose $M$ be a $R$ module, $R$ is a commutative ring, now if we take the filtration $\{M_n\}$, then it induces a topology on $M$ and the book says that the completion of $M$ is $\varprojlim (M/M_n)$ and also the another definition is $\hat{M}$ is the set of all equivalence classes of Cauchy sequences of $M$. My question is-
$1.$ How this two definitions are equivalent?
$2.$What is the main idea of Completion in A topological group, i.e. what is Cauchy Sequences here and what about the convergence?
If anyone can give me some idea I'll be benefited.
Thanks in advance.
EDIT: Is there any book or material which helps to easily understand these things?
