Let X be an infinite set. Show that adding or subtracting a single point does not change its cardinality.
I have a plan but need help writing the actual proof.
I need to show that it doesn't matter which point is removed, and then I can use the fact that X is in one-to-one correspondence with a proper subset to prove this.
Best Answer
Hint: show that there exists an injective map $\omega\to X$. Then it is possible to "hide" a single point by shifting $\omega$ up/down.