Altering an Infinite Set does not change cardinality 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.
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.
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:
$$|X- \{x\}| = |X|$$