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Let $G(X)$ be the set of all Cauchy seq. on $(X,d)$ and define for $(x_n), \ (y_n)$ the following relation $(x_n) \sim (y_n) \Leftrightarrow d(x_n,y_n) \to 0 \ (n \to \infty)$ $(n \in \mathbb{N})$

a) Prove that this is an eq. relation (done)

b) Let $\hat{X} = G(X)/\sim$ and prove that (done) $$ \hat{d}([(x_n)], [(y_n)] = \lim_{n \to \infty} d(x_n, y_n) \\ [(x_n)], [(y_n)] \in \hat{X} $$

c) For every $x \in X$ the seq. $(x,x,x,x,x, \dots) \in G(X)$. Define $\tilde{X} = \{\tilde{x} : x \in X, \ \tilde{x} = [(x,x,x,x,x \dots )] \}$. Prove $X$ and $\tilde{X}$ are isometric, and $\tilde{X}$ is dense in $\hat{X}$

For c) i defined $f: X \to \tilde{X}$ by $f(x) = [(x,x,x,x,x, \dots)]$, then $\hat{d}(f(x),f(y)) = \lim_{n \to \infty} d(x,y) = d(x,y)$. Now I have left to prove that $cl(\tilde{X}) = \hat{X}$, but I need help with this.

I decieded to include all information in the exercise, since they might be helpful.

Olba12
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1 Answers1

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For $y = [(y_n)_{n}] \in \hat{X}$ you have to find a sequence $(x_k)_k$ in $X$ such that $\tilde{d}(f(x_k),y) \to 0$ for $k \to \infty$. You already know that $\hat{d}(f(x_k),y) = \lim\limits_{n \to \infty} d(y_n,x_k)$, so you have to define $(x_k)$ such that $d(y_n,x_k)$ is small for large $n$ and $k$.

Now, since $(y_n)_n$ is a Cauchy sequence, you already know that $d(y_n,y_k)$ is small for large $n$, $k$. So you could just take $x_k = y_k$ so that $\hat{d}(f(x_k),y) = \lim\limits_{n \to \infty} d(y_n,y_k) $ which approaches $0$ for $k \to \infty$.

Note that I omitted some of the important details here, I just wanted to give you a rough idea on how this works.

Matthias Klupsch
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