if $A$ is infinite and $B$ is a finite subset of $A+.1667em-.1667em,$ then $A\setminus B$ is infinite

Theorem. If $A$ is an infinite set and $B$ is afinite subset of $A$, then $A\setminus B$ is infinite.

Proof. The proof is by contradiction. If $A\setminus B$would be finite, there would exist a $k\in \mathbb{N}$ and a bijection$f:\{1,\mathrm{\dots },k\}\to A\setminus B$. Since $B$ is finite, therealso exists a bijection $g:\{1,\mathrm{\dots },l\}\to B$. We can then definea mapping $h:\{1,\mathrm{\dots },k+l\}\to A$ by

$h(i)$$=$

Since $f$ and $g$ are bijections, $h$ is a bijection betweena finite subset of $\mathbb{N}$ and $A$. This is a contradictionsince $A$ is infinite. $\mathrm{\square }$