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.

Your reasoning is informally correct.  If you want to give a careful answer to this question you should demonstrate a bijection from $A \setminus \{x\}$ to $\mathbb{N}$.
You have a given bijection between $A$ and $\mathbb{N}$.  Can you use this as a stepping stone to get from $A \setminus \{x\}$ to $\mathbb{N}$?
