How to show if A is denumerable and $x\in A$ then $A-\{x\}$ is denumerable My thoughts:
If $A$ is denumerable then it has a bijection with $\mathbb{N}$
So therefore $A\rightarrow \mathbb{N}$. Then x is a single object in A and A is infinite.
So if a single object is removed from then $A$ is still infinite.

Let
$$f:A\to\Bbb N$$
be a bijection which sends $x\mapsto n$. Then the map
$$\begin{cases}f':\Bbb N\to\Bbb N\setminus\{n\} \\ k\mapsto k & k<n \\ k\mapsto k+1 & k\ge n\end{cases}$$
is a bijection.
Composing with the obvious restriction, $f\bigg |_{A\setminus\{x\}}$ gives us that
$$f'^{-1}\circ f\bigg|_{A\setminus\{x\}}:A\setminus\{x\}\to \Bbb N$$
is a bijection.
