elementary set theory - 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.

Answer

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:
$$|X- \{x\}| = |X|$$