Show that $C(X,\Bbb R)$ is not separable

Question

Let $X$ be a metric space whose points are the positive integers and whose metric ($d\colon X\times X\rightarrow \mathbb{R}$) is defined by $$d(x,y)=\dfrac{|x-y|}{1+|x-y|}.$$ Show that $C(X,\mathbb{R})$ is not separable.

Answer 1

Assuming supremum norm in $C(X,\mathbb{R}$), it should not be difficult to see that $C(X,\mathbb{R})(\text{with metric}\ |x-y|\ \text{in}\ X)\subseteq C(X,\mathbb{R})(\text{with metric}\ d(x,y)\ \text{in}\ X\ \text{as defined above})$. Now all the binary sequences which are the members of (your) $C(X,\mathbb{R})$, form an uncountable discrete space (any two elements are unit distance apart), which is not separable. So, $C(X,\mathbb{R})$ is not separable.