Is the space $(\mathbb{R}\setminus\mathbb{Q},d)$ separable with the metric $d(x,y)=\frac{|x-y|}{1+|x-y|}$?

Question

Is the space $(\mathbb{R}\setminus\mathbb{Q},d)$ separable with the metric $d(x,y)=\dfrac{|x-y|}{1+|x-y|}$?

Here is what I think: The set of irrational numbers are uncountable. Since we cannot approximate any element from that space with a countable family of sets, it follows that the space is not separable. However, I could not relate this to the given metric.

Another proposition to prove the separability of the space is to use the fact that $(X,d)$ is separable iff $\forall \varepsilon$ $\exists$ countable $\varepsilon-$net.

How can I dis/prove the separability of the metric space?

Answer 1

Hint:

First notice that $d$ induces the standard Euclidean topology on $\mathbb{R}\setminus \mathbb{Q}$.

Then consider the countable set $$\mathbb{Q} + \sqrt2 = \{q+\sqrt{2} : q \in \mathbb{Q}\}$$ and show that it is dense in $\mathbb{R}\setminus \mathbb{Q}$.