This is homework 2, due Monday September 26 at the beginning of lecture.
Let be a metric space.
- Show that if is defined by , then is also a metric on .
- Show that if is open in then it is open in , and viceversa.
Recall that is open iff it is a union of open balls. Use this to explain why it suffices to show that if is open in then for any there is an such that
and similarly, if is open in then for any there is a such that
In turn, explain why to show this it suffices to prove that for any and any there is a such that
and, similarly, for any there is a such that
Finally, prove this by showing that we can take (no matter what is) and similarly, find an appropriate that works for (again, independently of ).
- Illustrate the above in as accurately as possible.
- Suppose that a sequence converges to in and to in . Show that .
- Is it true that a sequence is Cauchy in iff it is Cauchy in ? (Give a proof or else exhibit a counterexample, with a proof that it is indeed a counterexample.)
- Show that any dense subset of has the same size as .