Let $(X,d)$ a metric space and $\sim$ a equivalence relation such that :
- $\forall x\in X$ : $[x]=\{y\in X \vert y \sim x \}$ is closed.
- If $[x] \neq [y]$ : $d([x],[y])=d(a,[y]), \forall a\in[x]$
Define in $\dfrac{X}{\sim}$ : $D([x],[y])=d([x],[y])$ what is a metric (I've proved it). Prove that if $(X,d)$ is complete then $(\dfrac{X}{\sim},D)$ is complete.
My attempt :
Let $\{[x_n]\}$ a cauchy sequence of $\dfrac{X}{\sim}$. Then for $\epsilon >0$ exists $n_0 \in \mathbb{N}$ such that :
$$ \forall m,n \geq n_0 \implies D([x_m],[x_n])< \dfrac{\epsilon }{2} \implies D([x_m],[x_{n_0}])< \dfrac{\epsilon }{2}$$
By the definition of $D$ exists $p_m \in [x_m] $ such that
$$ d(p_m,x_{n_0})< \dfrac{\epsilon }{2} \implies d(p_m,p_n)<\epsilon $$
$\{p_m\}_{m\geq n_0}$ is cauchy?