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.

Suppose that $f:\mathbb{N}\to A$ is a bijection, where $f(n)=x$ for some $n\in\mathbb{N}$.

Define $$ g(k)=\left\\{\begin{array}{} f(k)&\text{if }k\lt n\\\ f(k+1)&\text{if }k\ge n\\\ \end{array} \right. $$ Then $g:\mathbb{N}\to A-\\{x\\}$ is a bijection.