Let $$X_{r}=\{2^{r}(2s-1)-1:s=1,2,3,...\}.$$
Show that $X_{n} \cap X_{m}=\emptyset$ for all $n\ne m$ and also the union of $X_{i}$ $(i\in \mathbb N)$ is $2\mathbb N+1$.
NB: $n$ is fixed. For a fixed natural number $n=1$
$$X_{1}=\{2(2s-1)-1:s=1,2,3...\}$$ Also I assume here $\mathbb N=\{0,1,2,...\}$