Let $(X,d)$ be a metric space.
Let $C(X)$ be the set of all Cauchy seq. on $X$ and define for $(x_n)_{n \in \mathbb{N}}$ , $(y_n)_{n \in \mathbb{N}}$ the following relation
$$ (x_n)_{n \in \mathbb{N}} \sim (y_n)_{n \in \mathbb{N}} \Leftrightarrow d(x_n,y_n) \to 0 \ (n \to \infty) $$
Let $\hat{X} = C(X) / \sim $ and prove that
$$ \hat{d}\left([(x_n)_{n \in \mathbb{N}}], [(y_n)_{n \in \mathbb{N}}]\right) = \lim_{n \to \infty} d(x_n, y_n) \\ ([(x_n)_{n \in \mathbb{N}}, [(y_n)_{n \in \mathbb{N}}] \in \hat{X} $$
is a metric on $\hat{X}$
Ideally I would say
$$ \hat{d}\left([(x_n)_{n \in \mathbb{N}}], [(y_n)_{n \in \mathbb{N}}]\right) = \lim_{n \to \infty} d(x_n, y_n) \leq \lim_{n \to \infty} \left( d(x_n, z_n) + d(z_n, y_n) \right) = \lim_{n \to \infty} d(x_n, z_n) + \lim_{n \to \infty} d(z_n, y_n) = \hat{d}\left([(x_n)_{n \in \mathbb{N}}], [(z_n)_{n \in \mathbb{N}}]\right) + \hat{d}\left([(z_n)_{n \in \mathbb{N}}], [(y_n)_{n \in \mathbb{N}}]\right) $$ But I need help verifying the above steps, is it safe to assume that $\lim_{n \to \infty} d(x_n, y_n)$ exists?
EDIT
I realised it follows from that $\mathbb{R}$ is complete.