i think im pretty stuck in this class...

here's the problem:

Let (X,d) be a metric space and let S be the set of cauchy sequences in S. define a relation "~" in X by saying "{s_k} ~ {t_k}" to mean d(s_k,t_k) goes to 0 as k goes to infinite.

i did the first problem which is to show that "~" is an equivalence relation, and i can do thesecond problem says:

let X_tilda denote the set of equivalence classes of S and s_tilda denote the equivalence classes of s={s_k}k=1 to infinite. show that the function p(s_tilda, t_tilda)= lim (k approaches infinite) d(s_k,t_k), s_tilda, t_tilda in X_tilda defines a metric.

but i am stuck on the third/fourth/fifth part which is

3. show that (X_tilda, p) is complete

4. for x in X define x_tilda to be the equivalence class of constant sequence {x,x,....}. show that the function going from x to x_tilda is an isometry of X onto a dense subset of X_tilda (isometry means d(x,y)=p(x_tilda,y_tilda))

5. show that when Y is a completion of X, then the inclusion map X going to Y extends to an isometry of X_tilda onto Y (Note: Y is a completion of X if a complete metric space Y contains X as a dense subspace) according to hint given, this problem is supposed to show that completion of X is unique, up to isometry.

i think like many other topology questions, i'm having trouble visualizing the problem (especially #4 and #5), and for #3 i am aware of the definition of complete (which means that every cauchy sequence in the metric space converges), but i'm having hard time applying the definition to prove this.

thanks in advance for any inputs!