Prove the following set is uncountable I need some help with part (b) of the question. Would appreciate feedback on whether my solution to part (a) is correct too.
For each  ∈ ℝ, define  = { +  ∶  ∈ ℤ}. Let  = { ∶  ∈ ℝ}.
(a) Prove that  is countable for every  ∈ ℝ.
(b) Prove that  is uncountable.
You may use without proof the fact that a set  is countable if and only if there is a sequence
0, 1, 2, … ∈  in which every element of  appears.
For part (a), I proved that for all  ∈ ℝ,  can be written in a sequence defined as below:
c2i = x - i
c2i+1 = x + i + 1
i.e.  = {x, x + 1, x - 1, x + 2, x - 2, x + 3, x - 3, ....}
For part (b), however, I am stuck on proving .
I believe it is probably something to do with cardinality of Unions? Since  is just a Union of 1, 2, 3 ...
However, in my current syllabus, one theorem I am taught is that:
 "Let A,B be countable infinite sets. Then A U B is countable."
Thank you for taking the time to read this and I appreciate all feedback! Thank you

For $b)$ we have that $Ax$ is distinct between $A(0)$ and $A(1)$. Hence the cardinality is at least the same as the interval $(0,1)$, which we know is uncountable.
