Explain briefly what is meant by a metric space, and by a Cauchy sequence in a metric space.
A function is called a pseudometric on if it satisfies all the conditions for a metric except the requirement that implies . If is a pseudometric on , show that the binary relation on defined by is an equivalence relation, and that the function induces a metric on the set of equivalence classes.
Now let be a metric space. If and are Cauchy sequences in , show that the sequence whose th term is is a Cauchy sequence of real numbers. Deduce that the function defined by
is a pseudometric on the set of all Cauchy sequences in . Show also that there is an isometric embedding (that is, a distance-preserving mapping) , where is the equivalence relation on induced by the pseudometric as in the previous paragraph. Under what conditions on is bijective? Justify your answer.