(ii) Use (i) to prove that is also separable with respect to the metric .
Let have the maximum metric . Let be the set of all polynomials with rational coefficients on . Let be the set of all real polynomials on . Since is dense in , .
is countable. Proof: Let be the set of all polynomials of degree with rational coefficients. Then , where there are 's. Now define . is countable because it is a countable union of countable sets.
Let such that . Let . Since , there exists such that for . Let .
by the triangle inequality.
, making . We have . Furthermore, , so . Since is countable, is separable.
Let and be defined as above with the metric . We still have and is countable. Let such that . Let . Since , there exists such that . Notice . From (i), we have . Now notice, .
, and by the same argument as in (i), is separable.