We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and $$\lim_{N \to \infty} \frac{\#\{n \leq N : x_n \in [c,d]\}}{\#\{n \leq N : x_n \in [a,b]\}} = \frac{d-c}{b-a}$$ for all $[c,d] \subseteq [a,b]$, with $c < d$, (here $\#S$ is the number of elements of a finite set $S$).

We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $\mathbb{R}$" if $(x_n)_{n=1}^\infty$ is uniformly distributed in any $[a,b]$, with $a < b$.

I have a construction for a sequence uniformly distributed in $\mathbb{R}$. Are there in the literature examples of uniformly distributed sequence in $\mathbb{R}$? References?

Thank you all for your help.