Cauchy Sequences in a Metric Space: Constant Sequences, Convergence of Termwise Distances, the Null Relation, and Independence of Representatives

Let $(X,d)$ be a metric space and $\mathbb{N}$ the set of natural numbers; sequences are indexed by $\mathbb{N}$ as in Sequence in a Set, and limits of real sequences are those of Limit of a Sequence of Real Numbers. Let $\mathcal{C}(X)$ be the set of Cauchy sequences in $(X,d)$.

1. (Constant sequences)¶ For $a\in X$ let $\bar{a}$ be the sequence all of whose terms equal $a$. Then $\bar{a}\in\mathcal{C}(X)$.

2. (Distances)¶ For all $x=(x_{k})_{k\in\mathbb{N}}$ and $y=(y_{k})_{k\in\mathbb{N}}$ in $\mathcal{C}(X)$ the real sequence $(d(x_{k},y_{k}))_{k\in\mathbb{N}}$ converges; write $\delta(x,y)$ for its limit. For all $x,y,z\in\mathcal{C}(X)$ and $a,b\in X$,

3. (Null relation)¶ For $x,y\in\mathcal{C}(X)$ write $x\approx y$ if $\delta(x,y)=0$. For all $x,y,z\in\mathcal{C}(X)$: $x\approx x$; if $x\approx y$, then $y\approx x$; and if $x\approx y$ and $y\approx z$, then $x\approx z$. For $x\in\mathcal{C}(X)$ let $[x]=\{y\in\mathcal{C}(X):\ x\approx y\}$. Then, for $x,y\in\mathcal{C}(X)$, $[x]=[y]$ holds if and only if $x\approx y$.

4. (Representatives)¶ If $x,x',y,y'\in\mathcal{C}(X)$ satisfy $x\approx x'$ and $y\approx y'$, then $\delta(x,y)=\delta(x',y')$.