I am reading a famous book by Kolmogorov and Fomin.
Definition:
Given a metric space $R$ with closure $[R]$, a complete metric space $R^*$ is called a completion of $R$ if $R\subset R^*$ and $[R]=R^*$, i.e., if $R$ is a subset of $R^*$ everywhere dense in $R^*$.
There is the following theorem in this book:
The authors wrote:
The points of the sequence $\{x_n\}$ also belong to $R^{**}$, where they form a fundamental sequence (why?).
It is clear that $x^{**}$ is independent of the choice of the sequence $\{x_n\}$ converging to the point $x^*$ (why?).
Are my proofs of these facts right?
proof of 1.:
Since $x_n\to x^*$, $|x_{n'}-x_{n^{''}}|\leq |x_{n'}-x^*|+|x_{n^{''}}-x^*|<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$ for $n^{'}\ge N$ and $n^{''}\ge N$, where $N$ is some natural number.
So, $\{x_n\}$ is a fundamental sequence in $R^*$ and $R^{**}$.
proof of 2.:
Let $\{x_n\}$ and $\{x^{'}_n\}$ be two sequences which converge to the point $x^*$.
Let $\{x^{''}_n\}$ be a sequence such that $x^{''}_n=x_{\frac{n+1}{2}}$ if $n$ is odd and $x^{''}_n=x^{'}_{\frac{n}{2}}$ if $n$ is even.
Then, $x^{''}_n\to x^*$.
So, $\{x^{''}_n\}$ is a fundamental sequence in $R^*$ and $R^{**}$.
So, $\{x^{''}_n\}$ converges to some point $x^{**}$ in $R^{**}$.
Since $x^{''}_{2n-1}=x_n$ and $x^{''}_{2n}=x^{'}_n$, $\{x_n\}$ and $\{x^{'}_n\}$ are subsequences of $\{x^{''}_n\}$.
So, $x_n\to x^{**}$ and $x^{'}_n\to x^{**}$.