This is a homework problem for class credit and I would appreciate a hint.
Let $A$ be an infinite set and suppose $x, y$ and $z$ are points in $A$. Show that $A$ has the same cardinality as $A\setminus\{x,y,z\}$.
So far I am trying to solve this for the more general case, let $A$ be an infinite set and $B$ be a finite subset of $A$, show that $A$ has the same cardinality as $A\setminus B$. If $B$ is finite and $A$ is infinite, then $A\setminus B$ is infinite because if $A\setminus B$ were finite then $A = A\setminus B \cup B$, the union of two finite sets and hence $A$ would be finite. Using Cantor-Bernstein we know that if there is a one-to-one function from $A$ to $A\setminus B$ and a one-to-one function from $A\setminus B$ to $A$, then $A$ and $A\setminus B$ are equivalent. It is easy to define a function $f:A\setminus B \to A$ that is one-to-one, $f(x) = x$.
Now I need to define a function $g:A \to A\setminus B$ that is one-to-one. I have a feeling that this involves using the axiom of choice.
Is this the right strategy? Have I made any errors? And can anyone give me a hint on how to define a one to one function from $A$ to $A\setminus B$?