You can show that if you have two topological spaces and and if you consider the product space you will have for any and that and .
He's saying that if you consider giving the usual topology induced by the usual norm then the induced topology is the same as if the product topology (when both copies of the reals are endowed with the usual topology). That said, for the product topology it's easy to prove things like . So now since are both dense in you can conclude that is dense in . etc.