I've been doing some work over infinite sets and I was trying to establish a bijective mapping from $\mathbb{R}-\mathbb{Z}$ to $\mathbb{R}$ in order to prove that $|\mathbb{R}-\mathbb{Z}|=|\mathbb{R}|$.
Because $\mathbb{R}-\mathbb{Z}$ is an infinte set, by the Axiom of Choice I have guaranteed that $\exists\{a_k\}_{k\in\mathbb{N}} \subset \mathbb{R}-\mathbb{Z}$, so we define $f$ as follows: \begin{align}f:\mathbb{R}-\mathbb{Z} &\longrightarrow \mathbb{R}\\ x&\longmapsto f(x):=\begin{cases} 0, & \text{if $x=a_0$}& \\ k+1, & \text{if $x=a_{3k+1}$}& \\ -k-1, & \text{if $x=a_{3k+2}$}& \quad \forall k\in\mathbb{N} \\ a_k, & \text{if $x=a_{3k+3}$}& \\ x, & \text{if $x\neq a_k$} \end{cases} \end{align}
$f(a_{0})=0; \quad f(a_{3k+1})=\mathbb{Z}^{+}; \quad f(a_{3k+2})=\mathbb{Z}^{-}; \quad f(a_{3k+3})=\{a_k\}_{k\in\mathbb{N}}$
The function is trivially bijective.
Is this mapping okay? Should I correct some formal aspects?