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 the solution on part a, the idea is correct, but there is a typo in $c_{2i}=x-i$.
For part (b): $[0,1)$ can be imbedded in $\mathscr{C}$ via the map $x\mapsto Ax$.
Let $x,y\in[0,1)$ with $Ax=Ay$, then there must be some $k\in\mathbb{Z}$ with $x=y+k$. As $|x-y|<1$, thus $k=0$ and $x=y$.
This proves the injectivity.
$[0,1)$ is uncountable and so is $\mathscr{C}$.
