Good old Baby Rudin.

Let $X$ be a metric space.

a. Call two Cauchy sequences $\left\{p_n\right\},\left\{q_n\right\}$ in X equivalent if $$\underset{n\to \mathrm{\infty }}{lim}d(p_n,q_n)=0.$$ Prove that this is an equivalence relation.

b. Let $X^{\ast }$ be the set of all equivalence classes so obtained. If $P,Q\in X^{\ast },$ $\left\{p_n\right\}\in P,$ $\left\{q_n\right\}\in Q,$ define $$\mathrm{\Delta }(P,Q)=\underset{n\to \mathrm{\infty }}{lim}d(p_n,q_n).$$

1. Show that this limit exists for any $\left\{p_n\right\}$ and $\left\{q_n\right\}.$

2. Show that $\mathrm{\Delta }$ is well-defined.

c. Prove that $(X^{\ast },\mathrm{\Delta })$ is a complete metric space.

d. For each $p\in X,$ there is a Cauchy sequence all of whose terms are $p.$ Let $P_p$ be the element of $X^{\ast }$ containing this sequence. Prove that $\mathrm{\Delta }(P_p,P_q)=d(p,q)$ for all $p,q\in X.$ In other words, the mapping $\phi$ defined by $\phi \left(p\right)=P_p$ is an isometry taking $X$ into $X^{\ast }.$

e. Prove that $\phi \left(X\right)$ is dense in $X^{\ast },$ and that $\phi \left(X\right)=X^{\ast }$ if $X$ is complete. By (d), we may identify $X$ and $\phi \left(X\right)$ and thus regard $X$ as embedded in the complete metric space $X^{\ast }.$ We call $X^{\ast }$ the completion of X.