Is the metric space $(\mathbb{R}^\omega, d_f) $ separable?
Solution 1:
Yes: Let $x \in \mathbb{R}^\omega$, $\varepsilon > 0$. Choose $N \in \mathbb{N}$ with $\sum_{i=N+1}^\infty 1/2^i< \varepsilon/2$. Then choose $r_1,\dots, r_N \in \mathbb{Q}$ with $\sum_{i=1}^N \frac{2^{-i}|x_i-r_i|}{1+|x_i-r_i|} < \varepsilon/2$. Set $r=(r_1,\dots,r_N,0,0,0,\dots)$. Now $d_f(x,r)< \varepsilon$. This shows that the set of rational sequences which are eventually $0$ is dense. This set is countable.