I am trying to show that if $(M,d)$ is a metric space, then if $C$ is the set of all Cauchy sequences on M and $\sim$ is an equivalence relation on $C$ given by $(x_n) \sim (y_n)$ iff $\lim d(x_n, y_n)=0.$ Then the set $C/{\sim}$ is complete with the metric $d'(X,Y)= \lim d(X_n, Y_n)$ is complete.
Any ideas?