Explain briefly what is meant by a metric space, and by a Cauchy sequence in a metric space.

A function $d: X \times X \rightarrow \mathbb{R}$ is called a pseudometric on $X$ if it satisfies all the conditions for a metric except the requirement that $d(x, y)=0$ implies $x=y$. If $d$ is a pseudometric on $X$, show that the binary relation $R$ on $X$ defined by $x R y \Leftrightarrow d(x, y)=0$ is an equivalence relation, and that the function $d$ induces a metric on the set $X / R$ of equivalence classes.

Now let $(X, d)$ be a metric space. If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are Cauchy sequences in $X$, show that the sequence whose $n$th term is $d\left(x_{n}, y_{n}\right)$ is a Cauchy sequence of real numbers. Deduce that the function $\bar{d}$ defined by

$$\bar{d}\left(\left(x_{n}\right),\left(y_{n}\right)\right)=\lim _{n \rightarrow \infty} d\left(x_{n}, y_{n}\right)$$

is a pseudometric on the set $C$ of all Cauchy sequences in $X$. Show also that there is an isometric embedding (that is, a distance-preserving mapping) $X \rightarrow C / R$, where $R$ is the equivalence relation on $C$ induced by the pseudometric $\bar{d}$ as in the previous paragraph. Under what conditions on $X$ is $X \rightarrow C / R$ bijective? Justify your answer.