Altering an Infinite Set does not change cardinality

Question

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.

Answer 1

Let $X$ be the infinite set, $Y \subset X$ the set for which there's a bijection $Y \rightarrow X$ (which means $|Y|=|X|$), and $x$ some element in $X$.
Since there's at least one element "missing" in $Y$: $$|X|=|Y| \leq |X- \{x\}|$$ Using the same reasoning: $$|X- \{x\}| \leq |X|$$ Conclusion: $$|X- \{x\}| = |X|$$

Answer 2

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.

Answer 3

HINT: Given two elements $x,y\in X$ there is a permutation of $X$, $\pi$, such that $\pi(x)=y$ and $\pi(y)=x$.