Show that there exist a bijection between $ℤ$ and $A$

Question

Let $A$ be the set of points of the form $(s,0)$ (the second component is still zero) where $s∈B⊂ℝ$ and $B$ is an infinite discrete set.

My question: Show that there exist a bijection between $ℤ$ and $A$.

Answer 1

Which metric space are you working in? This is important to specify...

In any case, the proof is probably going to go like this: B is a discrete set, so (in the context of your metric space) B needs to be countably infinite, then the bijection between B and $\mathbb{Z}$ is trivial to find (we know that $\mathbb{Z}$ is in bijection with $\mathbb{N}$, and we know that B is in bijection with $\mathbb{Z}$, so there must exist a bijection between B and $\mathbb{Z}$). Then note there clearly is a bijection between $A$ and $B$ (this is trivial as well, prove it however you like), and so $A$ and $\mathbb{Z}$ are in bijection, and you're done.